Surroundings 74114

In a cylinder with a piston, there is nitrogen N2 with a volume of 5 liters, a temperature of 25 degrees Celsius, and a pressure of 200 kPa.
We very slowly compress the gas in the cylinder to half its volume while its temperature does not change.
Determine the resulting gas pressure and the heat that passes through the walls of the cylinder to the surroundings.

Correct answer:

p₂ =  400 kPa
E =  693.1472 J

Step-by-step explanation:

$$\begin{gathered} V_1=5\text{l}\to {\text{m}}^3=5:1000{\text{m}}^3=0.005{\text{m}}^3 \\ T=25°C\to K=25+273.16K=298.16K \\ {p}_1=200\text{kPa} \\ {V}_2=V_1/2=\frac{1}{200}/2=\frac{1}{200\cdot 2}=\frac{1}{400}=0.0025{\text{m}}^3 \\ \frac{p\cdot V}{t}=const \\ {p}_1{V}_1=p_2{V}_2 \\ {p}_2=p_1\cdot \frac{V_1}{V_2}=200\cdot \frac{\frac{1}{200}}{\frac{1}{400}}=400\text{kPa} \end{gathered}$$

$$\begin{gathered} R=8.31446261815324\doteq 8.3145\text{J/K/mol} \\ M=2.14\cdot 10^{-3}\doteq 0.0021\text{kg/mol} \\ {P}_1=p_1\cdot 1000=200\cdot 1000=200000\text{Pa} \\ m=\frac{P_1\cdot V_1\cdot M}{R\cdot T}=\frac{200000\cdot 0.005\cdot 0.0021}{8.3145\cdot 298.16}\doteq 0.0009\text{kg} \\ n=m/M=0.0009/0.0021\doteq 0.4034\text{mol} \\ E=-n\cdot R\cdot T\cdot \ln (V_2/V_1)=-0.4034\cdot 8.3145\cdot 298.16\cdot \ln (0.0025/0.005)=693.1472\text{J} \end{gathered}$$

Try another example