Nitrogen has a volume of 2.5 liters at a pressure of 100 KPa. Calculate how much the internal energy

Nitrogen has a volume of 2.5 liters at a pressure of 100 KPa. Calculate how much the internal energy of the gas has changed if, with a decrease in volume by 10 times, the pressure increased by 20 times.

Let’s translate all the values ​​from given to the SI system:
V = 2.5 l = 2.5 * 10 ^ -3 m³.
p = 100 kPa = 100 * 10 ^ 3 Pa.
The change in the internal energy of a diatomic gas is determined from the expression:
∆U = (5/2) * ν * R * ∆T
Mendeleev-Clapeyron equation
pV = (m / M) * R * T, where p is the gas pressure, V is the gas volume, m is the gas mass, M is the molar mass of the gas, R is the universal gas constant 8.31 J / (mol * K), T is the gas temperature.
m / M = ν, taking this into account we have:
pV = ν * R * T
Expression for the Mendeleev-Clapeyron equation before the process:
p1V1 = ν * R * T1
After the process:
p2V2 = ν * R * T2
Let’s find the difference:
p2V2-p1V1 = ν * R * T2-ν * R * T1 = ν * R * (T2- T1) = ν * R * ∆T
Let us substitute this in the formula for determining the change in the internal energy of the gas:
∆U = (5/2) * ν * R * ∆T
∆U = (5/2) * (p2V2-p1V1)
We have on condition:
V1 / V2 = 10, so: v2 = V1 / 10
p2 / p1 = 20, so: p2 = p1 * 20
With this in mind, we get:
∆U = (5/2) * (p1 * 20 * V1 / 10-p1 * V1)
Substitute the numerical values ​​and find the change in energy:
∆U = (5/2) * (p1 * 20 * V1 / 10-p1 * V1) = (5/2) * ((100 * 10 ^ 3) * 20 * (2.5 * 10 ^ -3) / 10- (100 * 10 ^ 3) * (2.5 * 10 ^ -3)) = 625 J.
Answer: the internal energy of the gas will change by 625 J.