Heat capacity. Her types. Relationship between heat capacities. Mayer's Law. Average and true specific heats. Heat capacity of a mixture of gases. Average heat capacity of a gas in the temperature range from m1 to m2 Average heat capacity of a substance
Is the amount of heat supplied to 1 kg of a substance when its temperature changes from T 1 to T 2 .
1.5.2. Heat capacity of gases
The heat capacity of gases depends on:
type of thermodynamic process (isochoric, isobaric, isothermal, etc.);
kind of gas, i.e. on the number of atoms in a molecule;
gas state parameters (pressure, temperature, etc.).
A) Influence of the type of thermodynamic process on the heat capacity of the gas
The amount of heat required to heat the same amount of gas in the same temperature range depends on the type of thermodynamic process performed by the gas.
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V isobaric process (R= const), heat is spent not only on heating the gas by the same amount as in the isochoric process, but also on performing work when the piston is raised with an area by a value (Fig. 1.2 b). The heat capacity of a gas in an isobaric process is indicated by the symbol with R .
Since, according to the condition, in both processes the value is the same, then in the isobaric process due to the work done by the gas, the value. Therefore, in the isobaric process, the heat capacity with R with υ .
According to Mayer's formula for ideal gas
or . (1.6)
B) Influence of the kind of gas on its heat capacity It is known from the molecular-kinetic theory of an ideal gas that
where is the number of translational and rotational degrees of freedom of motion of molecules of a given gas. Then
, a 
.
(1.7)
A monatomic gas has three translational degrees of freedom of movement of the molecule (Fig. 1.3 a), i.e. ...
Diatomic gas has three translational degrees of freedom of movement and two degrees of freedom of rotational motion of the molecule (Fig. 1.3 b), i.e. ... Similarly, it can be shown that for a triatomic gas.
Thus, the molar heat capacity of gases depends on the number of degrees of freedom of motion of molecules, i.e. on the number of atoms in the molecule, and the specific heat also depends on the molecular weight, since the value of the gas constant depends on it, which is different for different gases.
C) Influence of gas state parameters on its heat capacity
The heat capacity of an ideal gas depends only on temperature and increases with increasing T.
Monatomic gases are an exception because their heat capacity is practically independent of temperature.
The classical molecular kinetic theory of gases makes it possible to fairly accurately determine the heat capacity of monatomic ideal gases in a wide range of temperatures and the heat capacity of many diatomic (and even triatomic) gases at low temperatures.
But at temperatures significantly different from 0 ° C, the experimental values of the heat capacity of di- and polyatomic gases turn out to be significantly different from those predicted by the molecular-kinetic theory.
In heat engineering calculations, they usually use the experimental values of the heat capacity of gases, presented in the form of tables. In this case, the heat capacity determined in the experiment (at a given temperature) is called true heat capacity. And if the experiment measured the amount of heat q, which was spent on a significant increase in the temperature of 1 kg of gas from a certain temperature T 0 to temperature T, i.e. on T = T T 0, then the ratio
called middle heat capacity of the gas in a given temperature range.
Typically, in look-up tables, average heat capacities are given at the value T 0, corresponding to zero degrees Celsius.
Heat capacity real gas depends, in addition to temperature, also on pressure due to the influence of the forces of intermolecular interaction.
Heat capacity is a thermophysical characteristic that determines the ability of bodies to give or receive heat in order to change the body temperature. The ratio of the amount of heat supplied (or removed) in this process to the change in temperature is called the heat capacity of the body (system of bodies): C = dQ / dT, where is the elementary amount of heat; - elementary temperature change.
The heat capacity is numerically equal to the amount of heat that must be supplied to the system so that at given conditions increase its temperature by 1 degree. The unit of heat capacity is J / K.
Depending on the quantitative unit of the body to which heat is supplied in thermodynamics, one distinguishes between mass, volume and molar heat capacities.
Mass heat capacity is the heat capacity per unit mass of the working fluid, c = C / m
The unit of mass heat capacity measurement is J / (kg × K). The mass heat capacity is also called the specific heat capacity.
Volumetric heat capacity is the heat capacity per unit volume of the working fluid, where and are the volume and density of the body at normal physical conditions... C '= c / V = c p. The volumetric heat capacity is measured in J / (m 3 × K).
The molar heat capacity is the heat capacity related to the amount of the working fluid (gas) in moles, C m = C / n, where n is the amount of gas in moles.
The molar heat capacity is measured in J / (mol × K).
Mass and molar heat capacities are related by the following relationship:
The volumetric heat capacity of gases is expressed through molar as
Where m 3 / mol is the molar volume of gas under normal conditions.
Mayer's equation: С р - С v = R.
Considering that the heat capacity is not constant, but depends on temperature and other thermal parameters, distinguish between the true and average heat capacity. In particular, if one wants to emphasize the dependence of the heat capacity of the working fluid on temperature, then write it down as C (t), and the specific one as c (t). Usually, the true heat capacity is understood as the ratio of the elementary amount of heat that is imparted to a thermodynamic system in any process to the infinitesimal increase in the temperature of this system caused by the imparted heat. We will consider C (t) as the true heat capacity of the thermodynamic system at the temperature of the system equal to t 1, and c (t) as the true specific heat capacity of the working fluid at its temperature equal to t 2. Then the average specific heat of the working fluid when its temperature changes from t 1 to t 2 can be determined as
Usually, the tables give the average values of the heat capacity c av for different temperature ranges starting with t 1 = 0 0 C. Therefore, in all cases when the thermodynamic process takes place in the temperature range from t 1 to t 2, in which t 1 ≠ 0, the number the specific heat q of the process is determined using the tabular values of the average heat capacities c av as follows.
This is the amount of heat that must be communicated to the system to increase its temperature by 1 ( TO) in the absence of useful work and the constancy of the corresponding parameters.
If we take an individual substance as a system, then total heat capacity of the system equals the heat capacity of 1 mole of a substance () multiplied by the number of moles ().
Heat capacity can be specific or molar.
Specific heat is the amount of heat required to heat a unit mass of a substance by 1 hail(intense value).
Molar heat capacity is the amount of heat required to heat one mole of a substance per 1 hail.
Distinguish between true and average heat capacity.
In technology, the concept of average heat capacity is usually used.
Average is the heat capacity for a certain temperature range.
If a system containing an amount of a substance or a mass has been informed by the amount of heat, and the temperature of the system has increased from to, then the average specific or molar heat capacity can be calculated:
True molar heat capacity is the ratio of the infinitesimal amount of heat imparted by 1 mole of a substance at a certain temperature to the temperature increment that is observed in this case.
According to equation (19), heat capacity, like heat, is not a function of state. At constant pressure or volume, according to equations (11) and (12), heat, and, consequently, heat capacity acquire the properties of a function of state, that is, they become characteristic functions of the system. Thus, we obtain isochoric and isobaric heat capacities.
Isochoric heat capacity- the amount of heat that must be communicated to the system in order to increase the temperature by 1, if the process occurs at.
Isobaric heat capacity- the amount of heat that must be communicated to the system in order to increase the temperature by 1 at.
The heat capacity depends not only on the temperature, but also on the volume of the system, since there are interaction forces between the particles, which change when the distance between them changes, therefore, partial derivatives are used in equations (20) and (21).
The enthalpy of an ideal gas, like its internal energy, is only a function of temperature:
and in accordance with the Mendeleev-Clapeyron equation, then
Therefore, for an ideal gas in equations (20), (21), the partial derivatives can be replaced by total differentials:
From the joint solution of equations (23) and (24), taking into account (22), we obtain the equation of the relationship between and for an ideal gas.
By dividing the variables in equations (23) and (24), it is possible to calculate the change in internal energy and enthalpy when 1 mol of an ideal gas is heated from temperature to
If in the indicated temperature range the heat capacity can be considered constant, then as a result of integration we obtain:
Let's establish the relationship between the average and true heat capacity. The change in entropy, on the one hand, is expressed by equation (27), on the other,
Equating the right-hand sides of the equations and expressing the average heat capacity, we have:
A similar expression can be obtained for the average isochoric specific heat.
The heat capacity of most solid, liquid and gaseous substances increases with increasing temperature. The dependence of the heat capacity of solid, liquid and gaseous substances on temperature is expressed by an empirical equation of the form:
where a, b, c and - empirical coefficients calculated on the basis of experimental data on, and the coefficient refers to organic substances, and - to inorganic. Coefficient values for various substances given in the manual and are applicable only for the specified temperature range.
The heat capacity of an ideal gas is independent of temperature. According to the molecular kinetic theory, the heat capacity per degree of freedom is equal to (the degree of freedom is the number of independent types of motion into which the complex motion of a molecule can be decomposed). For a monatomic molecule, translational motion is characteristic, which can be decomposed into three components in accordance with three mutually perpendicular directions along three axes. Therefore, the isochoric heat capacity of a monatomic ideal gas is
Then the isobaric heat capacity of a monatomic ideal gas according to (25) is determined by the equation
Diatomic molecules of an ideal gas, in addition to three degrees of freedom of translational motion, also have 2 degrees of freedom of rotational motion. Hence.
Heat capacity is the ratio of the amount of heat δQ received by a substance with an infinitesimal change in its state in any process to the change in temperature dT of the substance (symbol C, unit J / K):
С (T) = δQ / dT
The heat capacity of a unit of mass (kg, g) is called the specific (unit J / (kg K) and J / (g K)), and the heat capacity of 1 mol of a substance is called the molar heat capacity (unit J / (mol K)).
Distinguish between true heat capacity.
С = δQ / dT
Average heat capacity.
Ĉ = Q / (T 2 - T 1)
Average and true heat capacities are related by the ratio
The amount of heat absorbed by a body when its state changes depends not only on the initial and final state of the body (in particular, on temperature), but also on the conditions for the transition between these states. Consequently, its heat capacity also depends on the heating conditions of the body.
In an isothermal process (T = const):
C T = δQ T / dT = ± ∞
In an adiabatic process (δQ = 0):
C Q = δQ / dT = 0
Heat capacity at constant volume, if the process is carried out at constant volume - isochoric heat capacity C V.
Heat capacity at constant pressure, if the process is carried out at constant pressure - isobaric heat capacity С P.
At V = const (isochoric process):
C V = δQ V / dT = (ϭQ / ϭT) V = (ϭU / ϭT) V
δQ V = dU = C V dT
At Р = const (isobaric process)%
C p = δQ p / dT = (ϭQ / ϭT) p = (ϭH / ϭT) p
The heat capacity at constant pressure C p is greater than the heat capacity at constant volume C V. When heated at constant pressure, part of the heat is used to produce the work of expansion, and part to increase the internal energy of the body; when heated at a constant volume, all the heat is spent on increasing the internal energy.
The relationship between C p and C V for any systems that can only do the work of the extension. According to the first law of thermodynamics%
δQ = dU + PdV
Internal energy is a function of external parameters and temperature.
dU = (ϭU / ϭT) V dT + (ϭU / ϭV) T dV
δQ = (ϭU / ϭT) V dT + [(ϭU / ϭV) T + P] dV
δQ / dT = (ϭU / ϭT) V + [(ϭU / ϭV) T + P] (dV / dT)
The dV / dT value (volume change with temperature change) is the ratio of the increments of independent variables, that is, the value is undefined if the nature of the process in which heat transfer occurs is not indicated.
If the process is isochoric (V = const), then dV = 0, dV / dT = 0
δQ V / dT = C V = (ϭU / ϭT) V
If the process is isobaric (P = const).
δQ P / dT = C p = C V + [(ϭU / ϭV) T + P] (dV / dT) P
For any simple systems, it is true:
C p - C v = [(ϭU / ϭV) T + P] (dV / dT) P
Solidification and boiling point of the solution. Cryoscopy and ebulioscopy. Determination of the molecular weight of the solute.
Crystallization temperature.
A solution, unlike a pure liquid, does not completely solidify at a constant temperature; at a temperature called the temperature of the onset of crystallization, crystals of the solvent begin to precipitate, and as the crystallization proceeds, the temperature of the solution decreases (therefore, the freezing point of the solution is always understood as the temperature of the onset of crystallization). The freezing of solutions can be characterized by the value of the decrease in the freezing point ΔТ deputy, equal to the difference between the freezing temperature of a pure solvent T ° deputy and the temperature of the onset of crystallization of the solution T deputy:
ΔT deputy = T ° deputy - T deputy
Solvent crystals are in equilibrium with the solution only when the saturated vapor pressure above the crystals and above the solution is the same. Since the vapor pressure of the solvent above the solution is always lower than that over the pure solvent, the temperature corresponding to this condition will always be lower than the freezing point of the pure solvent. In this case, the decrease in the freezing temperature of the solution ΔT deputy does not depend on the nature of the solute and is determined only by the ratio of the number of particles of the solvent and the solute.
Lowering the freezing point of dilute solutions
Lowering the freezing point of the solution ΔT deputy is directly proportional to the molar concentration of the solution:
ΔT deputy = Km
This equation is called the second Raoult's law. The proportionality coefficient K - the cryoscopic constant of the solvent - is determined by the nature of the solvent.
Boiling temperature.
The boiling point of solutions of a non-volatile substance is always higher than the boiling point of a pure solvent at the same pressure.
Any liquid - solvent or solution - boils at the temperature at which the saturated vapor pressure becomes equal to the external pressure.
Increasing the boiling point of dilute solutions
An increase in the boiling point of solutions of non-volatile substances ΔT k = T k - T ° k is proportional to a decrease in the saturated vapor pressure and, therefore, is directly proportional to the molar concentration of the solution. The proportionality coefficient E is the ebulioscopic constant of the solvent, which does not depend on the nature of the solute.
ΔT to = Em
Raoult's second law. A decrease in the freezing point and an increase in the boiling point of a dilute solution of a non-volatile substance is directly proportional to the molar concentration of the solution and does not depend on the nature of the solute. This law is valid only for infinitely dilute solutions.
Ebulioscopy- a method for determining molecular weights by increasing the boiling point of a solution. The boiling point of a solution is the temperature at which the vapor pressure above it becomes equal to the external pressure.
If the solute is not volatile, then the vapor above the solution consists of solvent molecules. Such a solution begins to boil at a higher temperature (T) compared to the boiling point of a pure solvent (T0). The difference between the boiling points of a solution and a pure solvent at a given constant pressure is called the rise in the boiling point of the solution. This value depends on the nature of the solvent and the concentration of the solute.
A liquid boils when the pressure of the saturated vapor above it is equal to the external pressure. When boiling, the liquid solution and the vapor are in equilibrium. If the solute is not volatile, the increase in the boiling point of the solution obeys the equation:
∆ isp H 1 is the enthalpy of evaporation of the solvent;
m 2 is the molality of the solution (the number of moles of the solute per 1 kg of solvent);
E - ebulioscopic constant, equal to the increase in the boiling point of a one-molar solution compared to the boiling point of a pure solvent. The value of E is determined by the properties of only the solvent, but not the solute.
Cryoscopy- a method for determining molecular weights by lowering the freezing point of a solution. When the solutions are cooled, they freeze. Freezing point - the temperature at which the first crystals of the solid phase are formed. If these crystals consist only of solvent molecules, then the freezing point of the solution (T) is always lower than the freezing point of the pure solvent (T pl). The difference between the freezing temperatures of the solvent and the solution is called the lowering of the freezing point of the solution.
The quantitative dependence of lowering the freezing point on the concentration of the solution is expressed by the following equation:
M 1 - molar mass solvent;
∆ pl H 1 is the enthalpy of melting of the solvent;
m 2 - molality of the solution;
K is a cryoscopic constant, depending on the properties of only the solvent, equal to a decrease in the freezing point of a solution with a molality of the substance dissolved in it, equal to unity.
Temperature dependence of the saturated vapor pressure of the solvent.
Lowering the freezing point and increasing the boiling point of solutions, their osmotic pressure do not depend on the nature of the dissolved substances. Such properties are called colligative. These properties depend on the nature of the solvent and the concentration of the solute. As a rule, colligative properties appear when two phases are in equilibrium, one of which contains a solvent and a solute, and the other contains only a solvent.
purpose of work
Experimentally determine the values of the average heat capacity of air in the temperature range from t 1 to t 2, establish the dependence of the heat capacity of air on temperature.
1. Determine the power spent on heating gas from t 1
before t 2 .
2. Record the air flow values at the specified time interval.
Laboratory Preparation Instructions
1. To work out the section of the course “Heat capacity” according to the recommended literature.
2. To get acquainted with this methodological manual.
3. Prepare protocols laboratory work, including the necessary theoretical material related to this work (calculation formulas, diagrams, graphs).
Theoretical introduction
Heat capacity- the most important thermophysical quantity, which is directly or indirectly included in all heat engineering calculations.
Heat capacity characterizes the thermophysical properties of a substance and depends on the molecular weight of the gas μ , temperature t, pressure R, the number of degrees of freedom of the molecule i, from the process in which heat is supplied or removed p = const, v =const... The heat capacity most significantly depends on the molecular weight of the gas μ ... So, for example, the heat capacity for some gases and solids is
Thus, the less μ , the less substance is contained in one kilomole and the more heat needs to be supplied to change the gas temperature by 1 K. That is why hydrogen is a more efficient coolant than, for example, air.
Numerically, heat capacity is defined as the amount of heat that must be brought to 1 kg(or 1 m 3), a substance to change its temperature by 1 K.
Since the amount of supplied heat dq depends on the nature of the process, then the heat capacity also depends on the nature of the process. One and the same system in different thermodynamic processes has different heat capacities - c p, c v, c n... Of greatest practical importance are c p and c v.
According to the molecular kinematic theory of gases (MKT) for a given process, the heat capacity depends only on the molecular weight. For example, the heat capacity c p and c v can be defined as
For air ( k = 1,4; R = 0,287 kj/(kg· TO))
kJ / kg
For a given ideal gas, the heat capacity depends only on temperature, i.e.
Heat capacity of the body in this process called the ratio of heat dq received by the body with an infinitely small change in its state to a change in body temperature by dt
True and average heat capacity
The true heat capacity of the working fluid is understood as:
The true heat capacity expresses the value of the heat capacity of the working fluid at a point at the given parameters.
The amount of heat transferred. expressed in terms of the true heat capacity, can be calculated by the equation
Distinguish:
Linear dependence of heat capacity on temperature
where a- heat capacity at t= 0 ° C;
b = tgα is the slope.
Nonlinear dependence of heat capacity on temperature.
For example, for oxygen, the equation is represented as
kJ / (kg K)
Under average heat capacity with t understand the ratio of the amount of heat in the 1-2 process to the corresponding change in temperature
kJ / (kg K)
Average heat capacity is calculated as:
Where t = t 1 + t 2 .
Calculation of heat by equation
difficult, since the tables give the value of the heat capacity. Therefore, the heat capacity in the range from t 1 to t 2 must be determined by the formula
.
If the temperature t 1 and t 2 is determined experimentally, then for m kg gas, the amount of heat transferred should be calculated using the equation
Average with t and with true heat capacities are related by the equation:
For most gases, the higher the temperature t, the higher the heat capacity c v, c p... Physically, this means that the more the gas is heated, the more difficult it is to heat it further.