A system consists of 3.2 mol of the monatomic gas neon (which may be treated as
an ideal gas). Initially the system has a temperature of 290 K and a volume of 1.2 m3. Heat is
added to this system at constant pressure until the volume triples, then more heat is added at
constant volume until the pressure doubles. Find the total heat added to the system.
To find the total heat added to the system, we need to calculate the heat added in each step and then sum them up.
Step 1: Heat added at constant pressure until the volume triples
In this step, the pressure remains constant, so we can use the formula for heat added at constant pressure:
Q1 = n * Cp * ΔT
Where:
Q1 is the heat added
n is the number of moles of the gas (given as 3.2 mol)
Cp is the molar specific heat capacity at constant pressure (for a monatomic ideal gas, Cp = 5/2 * R, where R is the gas constant)
ΔT is the change in temperature
First, we need to find the change in temperature. Since the pressure is constant, we can use the ideal gas law equation:
P1 * V1 / T1 = P2 * V2 / T2
Where:
P1 and V1 are the initial pressure and volume, respectively
T1 is the initial temperature (given as 290 K)
V2 is the final volume (three times the initial volume)
T2 is the final temperature
Plugging in the values:
P1 * V1 / T1 = P2 * V2 / T2
P1 * 1.2 / 290 = P2 * (3 * 1.2) / T2
Since the pressure doubles in the second step, we can say P2 = 2 * P1:
P1 * 1.2 / 290 = 2 * P1 * (3 * 1.2) / T2
1.2 / 290 = 6 * 1.2 / T2
T2 = (6 * 1.2 * 290) / 1.2
T2 = 1740 K
Now we can calculate the change in temperature:
ΔT = T2 - T1
ΔT = 1740 - 290
ΔT = 1450 K
Next, we can calculate the heat added in this step:
Q1 = n * Cp * ΔT
Q1 = 3.2 mol * (5/2 * R) * 1450 K
Q1 = 5800 * R
Step 2: Heat added at constant volume until the pressure doubles
In this step, the volume remains constant, so the heat added can be calculated using the formula for heat added at constant volume:
Q2 = n * Cv * ΔT
Where:
Q2 is the heat added
Cv is the molar specific heat capacity at constant volume (for a monatomic ideal gas, Cv = 3/2 * R)
ΔT is the change in temperature
Since the pressure doubles, we can say P2 = 2 * P1:
P1 * V1 / T1 = 2 * P1 * V1 / T2
T2 = 2 * T1
Now we can calculate the change in temperature:
ΔT = T2 - T1
ΔT = 2 * T1 - T1
ΔT = T1
Finally, we can calculate the heat added in this step:
Q2 = n * Cv * ΔT
Q2 = 3.2 mol * (3/2 * R) * 290 K
Q2 = 1392 * R
Total heat added:
To find the total heat added, we sum up Q1 and Q2:
Total heat added = Q1 + Q2
Total heat added = 5800 * R + 1392 * R
Total heat added = 7192 * R
Therefore, the total heat added to the system is 7192 times the gas constant (R).