A cylinder with a movable piston contains 14 moles of a monatomic ideal gas at a pressure of 2.26 × 105 Pa. The gas is initially at a temperature of 300 K. An electric heater adds 52600 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 CPCP = 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?
1) Q = nc change T
change T = 52600/(14*20.79)
2) V1/T1 = V2/T2
V2 = V1T2/T1
3) W = P(v2-v1)
To solve this problem, we 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.
1) We are given that the gas is initially at a temperature of 300 K and that an electric heater adds 52600 J of energy into the gas. We can use the equation:
ΔU = Q - W
Where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the gas. In this case, the work done is zero since the pressure remains constant. So the equation becomes:
ΔU = Q
Using the equation for the change in internal energy for a monatomic ideal gas:
ΔU = nCvΔT
Where n is the number of moles of the gas, Cv is the molar heat capacity at constant volume (Cv = CPCP = 20.79 J/K/mol), and ΔT is the change in temperature. Rearranging the equation, we can solve for ΔT:
ΔT = ΔU / (nCv)
Substituting the given values:
ΔT = 52600 J / (14 mol * 20.79 J/K/mol)
Simplifying:
ΔT = 52600 J / 290.66 J/K
ΔT ≈ 180.74 K
Therefore, the temperature of the gas after the energy is added is approximately 481 K.
2) Since the pressure remains constant, we can use the ideal gas law to find the change in volume. The ideal gas law states:
PV = nRT
Where P is the pressure, V is the volume, n is the number of moles of gas, R ≈ 8.314 J/(mol*K) is the ideal gas constant, and T is the temperature. Rearranging the equation, we can solve for V:
V = (nRT) / P
Since the pressure remains constant, we can write the equation as:
V1 = (nRT1) / P and V2 = (nRT2) / P
Where V1 is the initial volume, V2 is the final volume, T1 is the initial temperature, and T2 is the final temperature. Since we know the initial conditions and have calculated the final temperature, we can solve for the final volume:
V2 = (nRT2) / P
Substituting the values:
V2 = (14 mol * 8.314 J/(mol*K) * 481 K) / 2.26 × 105 Pa
Simplifying:
V2 ≈ 0.86 m^3
Therefore, the change in volume of the gas is approximately 0.86 m^3.
3) Since the pressure remains constant, the work done by the gas can be calculated using the equation:
W = PΔV
Where P is the pressure and ΔV is the change in volume. Substituting the given values:
W = 2.26 × 105 Pa * 0.86 m^3
W ≈ 1.94 × 105 J
Therefore, the work done by the gas during this process is approximately 1.94 × 105 J.
To answer these questions, we can use the ideal gas law, the first law of thermodynamics, and the definition of work.
1) To find the final temperature of the gas after the energy is added, we 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 first law of thermodynamics equation is:
ΔU = Q - W
Where
ΔU is the change in internal energy of the gas,
Q is the heat added to the gas, and
W is the work done by the gas.
In this case, the gas is being heated at constant pressure, so the work done by the gas is given by the equation:
W = PΔV
Where P is the pressure and ΔV is the change in volume.
We are given the initial pressure and the heat added to the gas. Since the pressure remains constant, we can ignore the work term (W = 0). Therefore, we can rearrange the first law of thermodynamics equation to solve for the change in internal energy (ΔU):
ΔU = Q
Substituting the given values, we have:
ΔU = 52600 J
The change in internal energy (ΔU) of an ideal monatomic gas is given by:
ΔU = nc_vΔT
where 'n' is the number of moles of gas, 'c_v' is the molar specific heat capacity at constant volume, and ΔT is the change in temperature.
We are given the number of moles of gas and the molar specific heat capacity at constant pressure (C_P) for a monatomic ideal gas. We can use the relationship between heat capacity ratios (C_P/C_V) for monatomic ideal gases to find c_v:
C_P/C_V = γ
where γ is the ratio of specific heat capacities and is equal to 5/3 for monatomic ideal gases.
Since C_P - C_V = R (universal gas constant), we can solve for C_V:
C_V = C_P - R = γR - R = (γ-1)R
So, for a monatomic ideal gas:
c_v = C_V / n = (γ-1)R / n
Substituting the values, we have:
ΔU = nc_vΔT
52600 = (14)((20.79)/(6.022 × 10^23))(ΔT)
Simplifying the equation, we can solve for ΔT:
ΔT = (52600 / (14)) * ((6.022 × 10^23) / (20.79))
As a result, we find the change in temperature (ΔT) of the gas after the energy is added.
2) To find the change in volume of the gas, we can use the ideal gas law:
PV = nRT
Where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.
Since the pressure remains constant, we can write the equation as:
V1 = nRT1 / P
And after the energy is added:
V2 = nRT2 / P
Subtracting the first equation from the second equation, we get:
ΔV = V2 - V1 = ((nRT2) / P) - ((nRT1) / P)
By substituting the given values, we find the change in volume (ΔV) of the gas.
3) To find the work done by the gas during this process, we can use the equation:
W = PΔV
Where P is the pressure and ΔV is the change in volume.
By substituting the given values, we find the work done by the gas.
Once you have the values for ΔT, ΔV, and W, you can answer the specific questions provided.