Three moles of an ideal monatomic gas are at a temperature of 396 K. Then 2438 J of heat is added to the gas, and 897 J of work is done on it. What is the final temperature of the gas?

To determine the final temperature of the gas, we can use the First Law of Thermodynamics, which states that the change in internal energy (ΔU) of a system is equal to the heat added (Q) minus the work done (W) on the system. Mathematically, it can be written as:

ΔU = Q - W

First, let's calculate the change in internal energy (ΔU). Since the gas is ideal and monatomic, it only has translational kinetic energy, which is given by the equation:

ΔU = (3/2) * n * R * ΔT

Where:
n = number of moles of gas = 3
R = Universal gas constant = 8.314 J/(mol * K) (approximately)
ΔT = change in temperature = Tf - Ti

The initial temperature (Ti) is 396 K. We want to find the final temperature (Tf).

Now, let's plug the values into the equation:

ΔU = (3/2) * 3 * 8.314 * ΔT

Next, we know that ΔU = Q - W, so we can substitute:

(3/2) * 3 * 8.314 * ΔT = 2438 - 897

Simplifying this equation:

67.086 * ΔT = 1541

Now, divide both sides by 67.086:

ΔT = 1541 / 67.086

ΔT is approximately equal to 23.005 K.

Finally, we can find the final temperature by adding the change in temperature to the initial temperature:

Tf = Ti + ΔT = 396 + 23.005 = 419.005 K

Therefore, the final temperature of the gas is approximately 419.005 K.