Five moles of an ideal monatomic gas with an initial temperature of$\mathbf{127}\mathbf{°}\mathit{C}$expand and, in the process, absorb 1500 J of heat and do 2100 J of work. What is the final temperature of the gas?

The given data can be listed below as:

• The heat absorbed during the process is, $Q=1500J$.
• The work done during the process is,$W=2100J$.
• The initial temperature is,$T_1=\left(127°C=273\right)K=400K$ .
• The number of moles is,$n=5mol$ .

The internal energy shows the dependency on the number of moles, the temperature difference of the gas, and the specific heat capacity. It can be measured in Joules.

Since, it is not mentioned if the pressure is constant or not, we can therefore use the first law of thermodynamics. Now as heat is added therefore work done is positive and the change in internal energy$∆U$ of the gas is expressed by:

$∆U=Q-W$

Substituting the given values in the equation, we get:

$$\begin{gathered} ∆U=1500J-2100J \\ =-600J \end{gathered}$$

Therefore, the change in internal energy is -600J .

Now the relationship between the change in internal energy and the change in temperature is given by:

$$\begin{gathered} ∆U=\frac{3}{2}nR∆Y \\ =\frac{3}{2}nR\left(T_2-T_1\right) \\ {T}_2=\frac{2∆U}{3nR}+T_1 \end{gathered}$$

Here, R is the Universal gas constant whose value is$8.314J/mol.K$ .

Now, substituting the values of $n,R,T,∆U$in the above equation and solve for$T_2$ , we get:

$$\begin{gathered} T_2=\frac{2\times \left(-600J\right)}{3\times 5mol\times 8.314J/mol.K}+400K \\ =\left(390.378K-273\right)°C \\ =117.378°C \end{gathered}$$

Therefore, the final temperature is$117.378°C$ .