To find the final temperature of the gas, we can use the equation:
Q = ΔU + W
where:
Q is the heat absorbed by the gas,
ΔU is the change in internal energy of the gas, and
W is the work done by the gas.
The change in internal energy, ΔU, for a monatomic ideal gas can be calculated using the equation:
ΔU = (3/2) * n * R * ΔT
where:
n is the number of moles of gas,
R is the ideal gas constant, and
ΔT is the change in temperature.
The work done by the gas, W, can be calculated using the equation:
W = -P * ΔV
where:
P is the pressure of the gas, and
ΔV is the change in volume.
In this case, we are given the heat absorbed (Q = 1300J) and the work done (W = 2040J). We need to find the final temperature (Tf).
Let's break down the problem step by step:
1. Convert the initial temperature from Celsius to Kelvin:
Ti = 123°C = (123 + 273.15) K
Ti = 396.15 K
2. Since we are given the number of moles (n = 5.00 mol), ideal gas constant (R = 8.3145 J/(mol*K)), and the heat absorbed (Q = 1300 J), we can rearrange the equation Q = ΔU + W to solve for the change in internal energy (ΔU).
ΔU = Q - W
ΔU = 1300 J - 2040 J
ΔU = -740 J
3. Now, substitute the known values into the equation for the change in internal energy (ΔU) and solve for the change in temperature (ΔT).
ΔU = (3/2) * n * R * ΔT
-740 J = (3/2) * 5.00 mol * 8.3145 J/(mol*K) * ΔT
-740 J = 62.1825 J/K * ΔT
ΔT = -740 J / 62.1825 J/K
ΔT ≈ -11.89 K
Note: A negative change in temperature indicates a decrease in temperature.
4. Finally, calculate the final temperature (Tf) using the initial temperature (Ti) and the change in temperature (ΔT).
Tf = Ti + ΔT
Tf = 396.15 K - 11.89 K
Tf ≈ 384.26 K
Therefore, the final temperature of the gas is approximately 384.26 K.