A cylinder with a movable piston contains 13.9 moles of a monatomic ideal gas at a pressure of 2.2 × 105 Pa. The gas is initially at a temperature of 300 K. An electric heater adds 5.2 × 104 J of energy into the gas while the piston moves in such a way that the pressure remains constant. It may help you to recall that CP = 20.79 J/K/mole for a monatomic ideal gas, and that the number of gas molecules is equal to Avagadros number (6.022 × 1023) times the number of moles of the gas.
1)What is the temperature of the gas after the energy is added?
2)What is the change in volume of the gas?
3)How much work is done by the gas during this process?
4)What is the change in the internal energy of the gas during this process?
To answer the given questions, we need to apply the laws of thermodynamics and use relevant equations. Let's tackle each question step by step:
1) What is the temperature of the gas after the energy is added?
To determine the new temperature, we can use the equation:
Q = n * C_P * ΔT
where Q is the energy added to the gas, n is the number of moles of the gas, C_P is the molar heat capacity at constant pressure, and ΔT is the change in temperature.
Rearranging the equation to solve for ΔT, we have:
ΔT = Q / (n * C_P)
Substituting the given values, we get:
ΔT = (5.2 × 10^4 J) / (13.9 moles * 20.79 J/K/mole)
ΔT ≈ 187.6 K
The change in temperature is approximately 187.6 K. To find the new temperature, we add the change in temperature to the initial temperature:
New Temperature = Initial Temperature + ΔT
New Temperature ≈ 300 K + 187.6 K
New Temperature ≈ 487.6 K
Therefore, the temperature of the gas after the energy is added is approximately 487.6 K.
2) What is the change in volume of the gas?
Since the pressure remains constant, we can use Boyle's Law, which states that the product of the initial volume and the initial pressure is equal to the product of the final volume and the final pressure:
P_initial * V_initial = P_final * V_final
We can rearrange the equation to solve for the change in volume:
ΔV = V_final - V_initial = (P_initial / P_final) * V_initial - V_initial
Substituting the given values, we get:
ΔV = (2.2 × 10^5 Pa / 2.2 × 10^5 Pa) * V_initial - V_initial
ΔV = 0
The change in volume is zero since the pressure remains constant. Therefore, there is no change in volume of the gas.
3) How much work is done by the gas during this process?
Since the pressure remains constant, the work done by the gas can be calculated using the equation:
W = P * ΔV
where W is the work done, P is the pressure, and ΔV is the change in volume.
As we found in the previous question, ΔV is zero. Therefore, the work done by the gas is also zero.
4) What is the change in the internal energy of the gas during this process?
The change in internal energy of the gas can be calculated using the equation:
ΔU = Q - W
where ΔU is the change in internal energy, Q is the energy added to the gas, and W is the work done by the gas.
Substituting the given values, we get:
ΔU = (5.2 × 10^4 J) - 0 J
ΔU = 5.2 × 10^4 J
Therefore, the change in internal energy of the gas during this process is 5.2 × 10^4 J.