How much heat had to be put in during the process to increase the internal energy of the gas by 1.50×10^4 J ?
Here is the given information and questions that I have already answered:
n=.450 R=8.3145
(Point A)
p=2*10^5 v=.010
(Point B)
p=5*10^5 v=.070
(Point C)
p=8*10^5 v=.070
Temp(A)=535 K
Temp(B)=9350 K
Temp(C)=15000 K
How much work was done by or on the gas in this process?
2.10*10^4 J
To calculate the amount of heat added, you can use 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.
The formula for the first law of thermodynamics is:
ΔU = Q - W
Given that the change in internal energy (ΔU) is 1.50×10^4 J and the work done (W) is 2.10×10^4 J (as calculated previously), we can rearrange the formula to solve for the heat added (Q):
Q = ΔU + W
Q = 1.50×10^4 J + 2.10×10^4 J
Q = 3.60×10^4 J
Therefore, the amount of heat that had to be put in during the process to increase the internal energy of the gas by 1.50×10^4 J is 3.60×10^4 J.
To determine the amount of heat that had to be put in during the process to increase the internal energy of the gas by 1.50×10^4 J, 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 (Q) added to the system minus the work (W) done by the system:
ΔU = Q - W
From the given information, we already know the work done by or on the gas is 2.10×10^4 J (which you have correctly calculated). Now we need to find the change in internal energy, ΔU.
The change in internal energy is given by the formula:
ΔU = Cᵥ * ΔT
where Cᵥ is the molar heat capacity at constant volume and ΔT is the change in temperature.
To calculate the change in internal energy, we need to find the molar heat capacity at constant volume (Cᵥ). For an ideal diatomic gas, the molar heat capacity at constant volume can be given by:
Cᵥ = (5/2) * R
where R is the ideal gas constant.
Given that R = 8.3145 J/(mol*K), we can calculate Cᵥ:
Cᵥ = (5/2) * 8.3145 = 20.78625 J/(mol*K)
Now, we can calculate the change in internal energy (ΔU) using the formula:
ΔU = Cᵥ * ΔT
For each of the given temperature changes, the ΔT can be calculated as follows:
ΔT(A) = Temp(B) - Temp(A) = 9350 K - 535 K = 8815 K
ΔT(B) = Temp(C) - Temp(B) = 15000 K - 9350 K = 5650 K
Substituting these values into the formula, we can calculate ΔU for each temperature change:
ΔU(A) = Cᵥ * ΔT(A) = 20.78625 J/(mol*K) * 8815 K = 183,060.53125 J
ΔU(B) = Cᵥ * ΔT(B) = 20.78625 J/(mol*K) * 5650 K = 117,502.0625 J
To find the total change in internal energy (ΔU) for the entire process, we sum up the individual changes in internal energy:
ΔU(total) = ΔU(A) + ΔU(B) = 183,060.53125 J + 117,502.0625 J = 300,562.59375 J
Now we can finally find the amount of heat (Q) that had to be put in during the process:
Q = ΔU + W
Q = 300,562.59375 J + 2.10×10^4 J
Q = 320,562.59375 J
Therefore, the amount of heat that had to be put in during the process to increase the internal energy of the gas by 1.50×10^4 J is approximately 320,562.59375 J.