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 find the final temperature of the gas, we can use the equation for the First Law of Thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system:

ΔU = Q - W

Where:
ΔU = change in internal energy
Q = heat added to the system
W = work done by the system

For an ideal monatomic gas, the change in internal energy (ΔU) is related to the number of moles (n) and the change in temperature (ΔT) by the equation:

ΔU = (3/2) nR ΔT

Where:
n = number of moles of gas
R = gas constant (8.314 J/(mol·K))

Let's calculate the change in internal energy (ΔU):

ΔU = (3/2) nR ΔT
ΔU = (3/2) (3) (8.314 J/(mol·K)) ΔT
ΔU = 35.21 ΔT

Now, let's substitute the given values into the equation for the First Law of Thermodynamics:

ΔU = Q - W
35.21 ΔT = 2438 J - 897 J

Now, let's solve for ΔT:

35.21 ΔT = 1541 J

Divide both sides of the equation by 35.21:

ΔT = 1541 J / 35.21

ΔT ≈ 43.76 K

Finally, to find the final temperature of the gas, we add the change in temperature to the initial temperature:

Final temperature = Initial temperature + ΔT
Final temperature = 396 K + 43.76 K

Final temperature ≈ 439.76 K

Therefore, the final temperature of the gas is approximately 439.76 K.