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Consider 1 kg of an ideal gas at 1 bar and 300 K contained in a rigid and perfectly insulated container. The specific heat of the gas at constant volume Cv is equal to 750 J.kg-1.K-1 . A stirrer performs 225 kJ of work on the gas. Assume that the container does not participate in the thermodynamic interaction. The final pressure of the gas will be ___________ bar (in integer).
Correct answer is '2'. Can you explain this answer?
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Consider 1 kg of an ideal gas at 1 bar and 300 K contained in a rigid ...
From first law,
TF = 600 K
Since, rigid container (volume = constant),
∵ p ∝T
p2 = 2 bar Hence, the final pressure of the gas will be 2 bar.
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Consider 1 kg of an ideal gas at 1 bar and 300 K contained in a rigid ...
Given data:
- Mass of the gas, m = 1 kg
- Pressure of the gas, P = 1 bar
- Temperature of the gas, T = 300 K
- Specific heat of the gas at constant volume, Cv = 750 J.kg^(-1).K^(-1)
- Work done on the gas, W = 225 kJ
Solution:
Step 1: Conversion of units
- 1 bar = 100,000 Pa (Pascal)
- 225 kJ = 225,000 J
Step 2: Calculation of initial internal energy
The internal energy of the gas can be calculated using the equation:
U = Cv * m * T
Substituting the given values, we get:
U_initial = 750 J.kg^(-1).K^(-1) * 1 kg * 300 K
= 225,000 J
Step 3: Calculation of final internal energy
The work done on the gas is given by the equation:
W = ΔU = U_final - U_initial
Where ΔU is the change in internal energy.
Rearranging the equation, we get:
U_final = W + U_initial
= 225,000 J + 225,000 J
= 450,000 J
Step 4: Calculation of final temperature
The final temperature of the gas can be calculated using the equation:
U_final = Cv * m * T_final
Rearranging the equation, we get:
T_final = U_final / (Cv * m)
= 450,000 J / (750 J.kg^(-1).K^(-1) * 1 kg)
= 600 K
Step 5: Calculation of final pressure
Since the container is rigid and perfectly insulated, the volume of the gas remains constant. Therefore, the final pressure can be calculated using the ideal gas law:
P * V = m * R * T_final
Where V is the volume of the gas and R is the specific gas constant.
Since the volume is constant, we can rewrite the equation as:
P_final = (m * R * T_final) / V
Since the container does not participate in the thermodynamic interaction, the volume remains constant. Therefore, we can simplify the equation as:
P_final = (m * R * T_final) / V
= (m * R * T_final) / (m * R)
= T_final
= 600 K
Converting the final pressure from Pascal to bar:
P_final = 600 K / 100,000 Pa/bar
= 6 bar
Therefore, the final pressure of the gas is 6 bar, which is not equal to the correct answer of 2 bar.
- Mass of the gas, m = 1 kg
- Pressure of the gas, P = 1 bar
- Temperature of the gas, T = 300 K
- Specific heat of the gas at constant volume, Cv = 750 J.kg^(-1).K^(-1)
- Work done on the gas, W = 225 kJ
Solution:
Step 1: Conversion of units
- 1 bar = 100,000 Pa (Pascal)
- 225 kJ = 225,000 J
Step 2: Calculation of initial internal energy
The internal energy of the gas can be calculated using the equation:
U = Cv * m * T
Substituting the given values, we get:
U_initial = 750 J.kg^(-1).K^(-1) * 1 kg * 300 K
= 225,000 J
Step 3: Calculation of final internal energy
The work done on the gas is given by the equation:
W = ΔU = U_final - U_initial
Where ΔU is the change in internal energy.
Rearranging the equation, we get:
U_final = W + U_initial
= 225,000 J + 225,000 J
= 450,000 J
Step 4: Calculation of final temperature
The final temperature of the gas can be calculated using the equation:
U_final = Cv * m * T_final
Rearranging the equation, we get:
T_final = U_final / (Cv * m)
= 450,000 J / (750 J.kg^(-1).K^(-1) * 1 kg)
= 600 K
Step 5: Calculation of final pressure
Since the container is rigid and perfectly insulated, the volume of the gas remains constant. Therefore, the final pressure can be calculated using the ideal gas law:
P * V = m * R * T_final
Where V is the volume of the gas and R is the specific gas constant.
Since the volume is constant, we can rewrite the equation as:
P_final = (m * R * T_final) / V
Since the container does not participate in the thermodynamic interaction, the volume remains constant. Therefore, we can simplify the equation as:
P_final = (m * R * T_final) / V
= (m * R * T_final) / (m * R)
= T_final
= 600 K
Converting the final pressure from Pascal to bar:
P_final = 600 K / 100,000 Pa/bar
= 6 bar
Therefore, the final pressure of the gas is 6 bar, which is not equal to the correct answer of 2 bar.
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Consider 1 kg of an ideal gas at 1 bar and 300 K contained in a rigid and perfectly insulated container. The specific heat of the gas at constant volume Cv is equal to 750 J.kg-1.K-1 . A stirrer performs 225 kJ of work on the gas. Assume that the container does not participate in the thermodynamic interaction. The final pressure of the gas will be ___________ bar (in integer).Correct answer is '2'. Can you explain this answer?
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