Question

During an adiabatic process the pressure of the gas is found to be proportional to fourth power of temperature. The ideal gas would be A. H2

B. He C. CH2 d micture of H2 and He

Answer 1

$\alpha, \beta, \gamma$ below are constants.

Ideal gas law:

$$PV=nRT, \\\space\\ T=\frac{PV}{nR}=\beta PV.$$

According to the condition:

$$P=\alpha T^4,$$

and since, as we established from Ideal gas law, the temperature is proportional to the product of pressure and time, we can write

$$T^4=(\beta PV)^4= \frac{P}{\alpha}.\\\space\\ P^3V^4=\frac{1}{\alpha \beta}.$$

Raise both sides of the equation to the power of 1/3:

$$PV^{4/3}=\frac{1}{(\alpha\beta)^3}.$$

An adiabatic process is described by

$$PV^\gamma=\text {const}.$$

Since the right part of the previous equation is a constant, comparing it with the adiabatic equation, we see that

$$\gamma=\frac43.$$

Express the adiabatic constant in terms of degree of freedom f:

$$\gamma=1+\frac{2}{f},\\\space\\ \frac43=1+\frac 2f,\\\space\\ f=6$$

degrees of freedom, which corresponds to

The answer is CH₂