Question

What will be the amount of heat absorbed by the gas if in a process, temperature and volume of one mole of an ideal monatomic gas varies according to the relation $VT=K$ , where $K$ is the constant and temperature of gas is increased by $\mathrm{\Delta }T$ ? It is given that $R$ is the gas constant.
A. ${\displaystyle \frac{1}{2}}R\mathrm{\Delta }T$
B. ${\displaystyle \frac{3}{2}}R\mathrm{\Delta }T$
C. ${\displaystyle \frac{1}{2}}KR\mathrm{\Delta }T$
D. ${\displaystyle \frac{2K}{3}}\mathrm{\Delta }T$

Answer 1

Hint: Use the ideal gas law. Find out the type of process and use its expression of molar specific heat capacity. Then, use the relation between heat and temperature when there is no work done.

Complete step by step answer:
According to the question, it is given that the ideal monatomic gas varies according to the relation –
$VT=K\cdots \left(1\right)$
Now, we have to use the ideal gas law, so you must know something about it.
Ideal gas law can be defined as the equation of state of hypothetical ideal gas. It is also known as the general gas equation. It is a good approximation of gases under many conditions but it has some limitations. The ideal gas law can be written in mathematical form as –
$PV=nRT$
where, $P$ is the pressure, $V$ is the volume ,$T$ is the temperature ,$n$ is the amount of substance, $R$ is the ideal gas constant.

So, this equation can be written as –
$T={\displaystyle \frac{PV}{nR}}\cdots \left(2\right)$
Putting the value of $T$ from equation $\left(2\right)$ in equation $\left(1\right)$, we get;
$$\begin{gathered} V\left[{\displaystyle \frac{PV}{nR}}\right]=K \\ \Rightarrow P{V}^2=nRK \end{gathered}$$
We know that, $nR$ is also the constant value, $\therefore nRK=K$
So, $P{V}^2=K\cdots \left(3\right)$
From equation $\left(3\right)$, we can conclude that this is the polytropic process. Polytropic process can be defined as the thermodynamic process which obeys the relation, $P{V}^n=K$ , where $K$ is the constant.
Now, using the expression of molar specific heat capacity.
$C={\displaystyle \frac{R}{1-x}}+C_v\cdots \left(4\right)$
Because this process follows polytropic process, $x=2$ and $C_v={\displaystyle \frac{3R}{2}}$
Putting these values in equation $\left(4\right)$, we get –
$$\begin{gathered} C={\displaystyle \frac{R}{1-2}}+{\displaystyle \frac{3R}{2}} \\ \Rightarrow C={\displaystyle \frac{3R}{2}}-R \\ \Rightarrow C={\displaystyle \frac{R}{2}} \end{gathered}$$
Now, using the relationship between heat and change in temperature when there is no work done –
$\mathrm{\Delta }Q=nC\mathrm{\Delta }T$
Putting the values in their respective places –
$\mathrm{\Delta }Q={\displaystyle \frac{R}{2}}\mathrm{\Delta }T$
Hence, this is the amount of heat absorbed by the gas.

Therefore, the correct option is A.

Note:To assume $P{V}^2=K$ as the equation of the polytropic process we have to assume $nR$ as constant. It was given in question that $R$ is the gas constant.In this question we used ideal gas law so, it is was first stated as the combination of empirical Boyle’s Law, Charles’s Law, Avogadro’s Law and Gay – Lussac’s Law.