It is customary to describe many gases in physics within the framework of an ideal model, with the help of which one can easily determine the basic thermodynamic characteristics of a system. In this article, we present the derivation of the Mayer formula for an ideal gas and show how it can be used to solve the problem.
The ideal gas and its equation
To understand the conclusion of the Mayer formula, one should become more familiar with the ideal gas model. According to her, a thermodynamic system is a collection of particles that do not have size (are material points). Their kinetic energy is the only component of the internal energy of the system (the potential energy of interactions is zero), and the particle velocities obey the classical Maxwell-Boltzmann distribution.
The described model is well suited for many real gaseous systems consisting of chemically inert atoms and molecules and having low pressures and temperatures of several hundred kelvin.
An ideal gas is described by a universal expression called the Clapeyron-Mendeleev law. It is given below:
P * V = n * R * T.
Here P, V, T are three thermodynamic quantities: pressure, volume, and temperature. Parameters n and R are the amount of substance in the system and constant for all gases, respectively.
Internal energy and isochoric process
According to the definition, the internal energy U of an arbitrary system is equal to the sum of the potential and kinetic energies of its elements. It was noted above that in an ideal model, gas molecules and atoms do not interact with each other, which means that the internal energy is determined exclusively by the kinetic component.
From the kinetic theory of gases, there follows an equality connecting the average kinetic energy of a particle with the absolute temperature in the system:
m * v 2/2 = z / 2 * k B * T.
Where m is the mass of one particle, k B is the Boltzmann constant, v is the mean square velocity, z is the number of degrees of freedom. If both parts of the equality are multiplied by the number N of particles in the system, then we obtain the expression for the internal energy U:
U = z / 2 * N * k B * T = z / 2 * n * R * T.
When writing this expression, we used the following equalities:
n = N / N A ; R = k B * N A.
Now we consider the question of determining the internal energy of a gas from the point of view of thermodynamics. We turn to the isochoric process. As a result of it, all the heat supplied goes to the heating of the system, since the volume remains constant, and the gas work is zero, that is:
dU = Q.
In turn, the change in the value of U can be written as follows:
dU = C V * dT.
Where C V is the heat capacity of the system at a constant volume. This value shows how much energy in joules needs to be spent to heat the system by 1 Kelvin. Comparing this expression with the formula for U, which is obtained from kinetic theory, we arrive at the equality:
C V = z / 2 * n * R.
Instead of heat capacity, C V is often used molar isochoric heat capacity, that is, the same value, only for 1 mole of gas:
C V = z / 2 * R.
Mayer formula for specific heat
To obtain the named formula, an isobaric process should be considered from the point of view of thermodynamics. By it is meant the transition of the system between two equilibrium states, in which the pressure does not change. Suppose that the system contains 1 mole of gas, then the first law of thermodynamics for the isobaric process will take the form:
Q = dU + P * dV.
Now we introduce the concept of isobaric heat capacity C P. It is understood as the amount of heat that should be reported to the system so that it is heated by 1 Kelvin, and its pressure does not change. This definition allows us to rewrite the equality above in the form:
C P * dT = dU + P * dV.
If we recall the equation for 1 mole of ideal gas and the expression for the change in internal energy, then this equality can be written as follows:
C P * dT = C V * dT + R * dT =>
C P = C V + R.
This expression is called the Mayer formula for ideal gas. It shows that the isobaric heat capacity is always greater than the isochoric one by the value of the gas constant for 1 mole of gas. The value of the gas constant R also takes on a specific physical meaning - this is the work that one mole of gas does during its isobaric expansion when heated by one Kelvin.
The task of determining the heat capacity of air
It is necessary to calculate the molar heat capacities C P and C V for air, assuming it to be an ideal gas.
It is not difficult to solve this problem, if we recall that the isochoric molar heat capacity is equal to:
C V = z / 2 * R.
Then, in accordance with the Mayer formula, the molar quantity C P will be equal to:
C P = C V + R = (2 + z) / 2 * R.
Since air is 99% formed by diatomic molecules O 2 and N 2 , then z is equal to 5. Substituting this value and the value R = 8.314 J / (K * mol) in the formulas, we obtain the answers to the problem: C V = 20 , 8 J / (K * mol), C P = 29.1 J / (K * mol).