91影视

The change in internal energy with regard to temperature is what is meant by the definition of the heat capacity at constant volume.

Heat energy at constant volume is given by

$\mathit{Q}\mathbf{=}\mathit{n}{\mathit{C}}_{\mathbf{p}}\mathbf{鈭�}\mathit{T}$

$\mathit{Q}\mathbf{=}\mathit{n}{\mathit{C}}_{\mathbf{p}}\left(T_b-T_a\right)$ 鈥�.. (1)

Here, Q is the amount of heat, n is the number of moles, $C_P$ is the heat capacity at constant pressure, $鈭�T$is the change in temperature.

Consider the given data as below.

The amount heat, $Q=-2.5脳{10}^{4}J$

Temperature, $T_2=283K$

Temperature, $T_1=323K$

T The heat capacity at constant volume, $C_V=20.76J/kg.K$

The heat capacity at constant pressure, $C_P=29.07J/mol.K$

Rearrange equation (1) for the number of moles as below.

$$\begin{gathered} n=\frac{Q}{C_P鈭�T} \\ =\frac{Q}{C_P\left(T_2-T_1\right)} \\ =\frac{-2.5脳{10}^4}{29.07脳\left(283-323\right)} \\ =21.5moles \end{gathered}$$

Hence, the number of moles, $n=21.5moles$

The change in internal energy is given by

$\mathbf{鈭�}\mathit{U}\mathbf{=}\mathit{n}{\mathit{C}}_{\mathbf{V}}\mathbf{鈭�}\mathit{T}$

Here, is the heat capacity at constant volume.

Therefore,

$$\begin{gathered} 鈭�U=n{C}_V鈭�T \\ =21.5脳20.76脳\left(283-323\right) \\ =-17.9KJ \end{gathered}$$

The internal energy decreases as change is negative.

According to the first law of thermodynamics,

$\mathit{W}\mathbf{=}\mathit{Q}\mathbf{-}\mathbf{鈭�}\mathit{U}$

Here, is the amount of work done by the gas, Q is the amount of heat, and $\mathbf{鈭�}\mathit{U}$is the change in internal energy.

As given

The amount of heat, $Q=2.5脳{10}^{4}J$

The change in internal energy. $鈭�U=-17.9KJ$

Work done is given by

$$\begin{gathered} W=-2.5脳{10}^4-\left(-17.9脳{10}^3\right) \\ =-7.1KJ \end{gathered}$$

Since, the work done is negative, work is done on the gas.

Now, since the volume is constant, the work done would be zero. According to the first law of thermodynamics

$$\begin{gathered} Q=W+鈭�U \\ =0+鈭�U \\ =-17.9KJ \end{gathered}$$

The heat liberated at constant volume is less than at constant pressure.