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The molar heat capacity of a gas at constant volume is \[{C_v}\]. If n mole of the gas undergo $\Delta T$ change in temperature, its internal energy will change by \[n{C_{v\;}}\] $\Delta T$
(A) Only if the change of temperature occurs at constant volume
(B) Only if the change of temperature occurs at constant pressure
(C) In any process which in not adiabatic
(D) In any process

Last updated date: 17th Jul 2024
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Hint: Use the first law of thermodynamics which states that, if the quantity of heat supplied to the system is capable of doing work, then the quantity of heat absorbed by the system is equal to the sum of the external work done by the system, and the increase in the internal energy of the system. Mathematically,
 $dQ = dW + dU$

 Complete step by step solution
According to the first law of thermodynamics,
 $dQ = dW + dU$ ……(i)
Where, dQ = Amount of heat added to the system.
     dW = External work done by the system.
     dU = Change in internal energy of the system.
Now, we know that specific heat of a gas at constant volume \[\left( {{C_v}} \right)\] is defined as the amount of heat required to raise the temperature of 1g gas through \[1^\circ C\] keeping the volume of the gas constant.
 ${C_v} = {\left( {\dfrac{{dQ}}{{dt}}} \right)_v} = {\left( {\dfrac{{dU}}{{dt}}} \right)_v}$ ……(ii)
Again, \[dW = PdV\] ……(iii)
Where, P = Pressure
     dV = Change in volume
As volume is constant,
 $dV = 0$
 $\therefore dW = 0$, (From equation (iii))
So, equation (i) becomes
 $dQ = dU$
Again, using equation (ii), we get
 $dU = {C_v}dT$
For n mole of gas,
 $dU = n{C_v}dT$

 This can only happen if the temperature change occurred at constant volume. Therefore correct option is A

 Note: In thermodynamics, state function is the property whose value does not depend on the path taken by the system to reach a specific value. For example, if a system changes from state 1 to state 2 then the value of dU will depend on the value of dT at state 1 and 2 but not on the path taken to reach the desired result.