Question

Two systems have the following equations of state, where

R is the gas constant per mole. The mole numbers are
​N1=3
N2 =5. The initial temperatures are T1 =175 K andT2​=400 K. What is the temperature once equilibrium is reached (in Kelvin)?

Answer 1

To find the equilibrium temperature of two systems, we balance the heat transfer equation using the initial temperatures and mole numbers. The calculated equilibrium temperature is approximately 316 K.

Step-by-step explanation:

The question asks to calculate the final equilibrium temperature when two systems with different initial temperatures and mole numbers reach thermal equilibrium.

To find the final temperature, we can use the principle of conservation of energy, because the total heat gained by the system that cools down must equal the total heat lost by the system that warms up. The equation for heat absorbed or released is Q = nCpΔT, where n is the number of moles, Cp is the molar heat capacity at constant pressure, and ΔT is the change in temperature.

Let's denote the two systems with subscripts 1 and 2, for system 1 and system 2, respectively. System 1 has N1 = 3 moles and an initial temperature T1 = 175 K. System 2 has N2 = 5 moles and an initial temperature T2 = 400 K. At equilibrium, both systems reach the same final temperature Teq.

Since energy is conserved, the heat lost by system 2 must be the heat gained by system 1:

N1 · Cp · (Teq - T1) = N2 · Cp · (T2 - Teq)

We can cancel Cp since it's the same for both systems (assuming ideal gas behavior) and solve for Teq.

3(Teq - 175) = 5(400 - Teq)

3Teq - 525 = 2000 - 5Teq

8Teq = 2525

Teq = 2525 / 8

Teq = 315.625 K

Therefore, the equilibrium temperature is approximately 316 K.