Question

In the process $P{{V}^{2}}=k$, if the temperature of gas is increased then
(A) Change in the internal energy of the gas is positive
(B) Work done by the gas is positive
(C) Heat is given to the gas
(D) Heat is taken out from the gas.

Answer 1

Hint: We know in a thermodynamic process there are mainly three main variables, pressure, volume and temperature. If we know any of the two variables then the third variable can be found out by using any of the thermodynamic equations. There are three laws which give us information about the thermodynamics process.

Complete answer:
Given process in the question is a polytropic process. The first law of thermodynamics written in mathematical form is $dQ=dU+dW$, where $dQ$ is the heat given to the system, $dU$ is the change in the internal energy of the system and $dW$ is the work done either by the system or on the system.

Also, change in the internal energy of the system is given by $U=n{{C}_{v}}dT$, now it is mentioned in the system that temperature of the gas is increased, so the change in the internal energy is positive. So, options A are correct.

Also, for a polytropic process, work done is given
$W=\dfrac{{{P}_{1}}{{V}_{1}}-{{P}_{2}}{{V}_{2}}}{n-1}$
So this will come out to be negative.Hence option B is incorrect. Heat is eventually given to the system because internal energy is already positive. So, option C is also correct and option D is wrong.

Hence, correct options are A and C.

Note: Polytropic processes are reversible processes. In these kinds of processes, the expression relating the properties of the system throughout the process is called the polytropic path. There is a term called polytropic index, which characterizes the process kind.The given equation in the question is $P{{V}^{2}}=k$. The power of the V is called the index of the process.