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In an adiabatic process, there is no heat exchange, hence, $Q=0$.

The equation $\mathrm dU=nC_V\,\mathrm dT$ is valid for every type of process given the gas is ideal, where $C_V$ is heat at constant volume required to raise the temperature of a system by 1°, q. We know that $Q=0$ for an adiabatic process, So $\mathrm dU$ = Work(adiabatic) And we define, $W$ (for an adiabatic process)$=nC_V\,\mathrm dT$ But how can $C_V$ have a numerical value for an adiabatic process, there is no heat exchange, no matter how much heat we provide, temperature of system would not increase, shouldn’t $C_V=0$? But then the equation won’t be valid.

Is the equation , $\mathrm dU=nC_V\,\mathrm dT$ even valid for an adiabatic process?

I don’t get it, where am I wrong? Please help me out, it’s my first time on here, apologies for any error.

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Consider a closed system for which $\mathrm{d}n=0$.

Then, the definition of $C_v$ is given by $\left(\dfrac{\partial U}{\partial T}\right)_V$.

$C_v$ is an intrinsic property of any material and it doesn't depend on the process involved.

You could interpret it as following:

If heat was provided at constant volume, it would raise the temperature of the system, and $C_v$ is a measure of this phenomenon.

In an adiabatic process, the work is still equal to $\int_{T_1}^{T_2}C_v\mathrm{d}T$. The temperature is changing in adiabatic processes because of the work and not the heat, which can both change the temperature of the system.

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