Question

An ideal gas is taken through a process where $dQ = dU - dW$ . The specific heat of the ideal gas in this process is ( $dQ$ = heat supplied to the gas, $dU$ = change in internal energy, $dW$ = work done by gas)
(A) ${C_v}$
(B) ${C_v} + \dfrac{3}{2}R$
(C) ${C_v} - R$
(D) The process is impossible

Answer 1

Hint : In the given question we have to find the specific heat of the ideal gas in the mentioned process. To find the required solution we must know the laws of thermodynamics, first law is required in this question and then will apply the given conditions in the equation.

Complete Step By Step Answer:
The first law of thermodynamics is based on the law of conservation of energy. The energy law states that energy can neither be created nor be destroyed but can only be transferred. It distinguishes two kinds of transfer of energy, as heat and thermodynamic work, relating them to a function of a body’s internal energy.
When a system changes from a given initial state to a given final state, the heat supplied and the work done depend on the nature of the process. Mathematically it is represented as
$\vartriangle U = Q - W$
$U$ denotes change in system
$Q$ is the quantity of energy supplied
$W$ is thermodynamic work done
It can also be written as
$dU = dQ - dW$
For the system the heat supplied is
$dQ = dU - dW$
According to first law of thermodynamics-
$dU = dQ - dW$
Then, heat supplied will be
$dQ = dU + dW$
Since in a system energy is always conserved, therefore we will equation these two equation of heat capacity
$dU + dW = dU - dW$
$dW = 0$
When heat supplied is zero
$dU = dQ$
This means that when heat is supplied to the system when work is zero, the internal energy equals the heat supplied. Therefore specific heat capacity ${C_v}$ will be equal the heat supplied or the heat capacity $C$
$C = {C_v}$ .

Note :
When there is no transfer of energy as heat or as work which means work is zero, heat supplied is zero, then the internal energy will be zero. This is known as the isolated system. The internal energy depends only on the initial and final stage.