A Cylinder is filled with 1.20 mol of He(g) at 25 C under ambient pressure of 0.966 atm. The gas in the cylinder is then heated with 1.430 KJ of heat, and the piston is raised by the expanding gas under the constant ambient pressure. Calculate the following after expansion.
a. final temperature or gas
b. work done by the gas
c. change in internal energy of the gas
d. final volume of gas
To solve this problem, we can use the ideal gas law and the first law of thermodynamics.
First, let's find the initial volume of the gas using the ideal gas law:
PV = nRT
Where:
P = pressure (0.966 atm)
V = volume (unknown)
n = moles of gas (1.20 mol)
R = gas constant (0.0821 L·atm/(mol·K))
T = temperature (25°C = 298 K).
Solving for V, we have:
V = nRT/P
V = (1.20 mol)(0.0821 L·atm/(mol·K))(298 K)/(0.966 atm)
V ≈ 29.29 L
(a) Final temperature of the gas:
To find the final temperature of the gas after expansion, we can use the formula for the first law of thermodynamics:
∆U = Q - W
Where:
∆U = change in internal energy (unknown)
Q = heat added to the gas (1.430 kJ)
W = work done by the gas (unknown).
Since the problem states that the expansion is under constant ambient pressure, we can assume that ∆U = Q - W. Rearranging the formula, we have:
W = Q - ∆U
W = 1.430 kJ - ∆U
(b) Work done by the gas:
We know that work done on or by a gas during a process at constant pressure is given by:
W = P∆V
Where:
P = pressure (0.966 atm)
∆V = change in volume (unknown).
We need to find ∆V, which is the difference between the final and initial volumes:
∆V = Vf - Vi
∆V = Vf - 29.29 L
The problem indicates that the piston is raised, so the gas expands and does work on its surroundings. Therefore, the work done by the gas is negative (-W). Substituting these values into the equation, we have:
-W = (0.966 atm)(∆V)
W = -(0.966 atm)(∆V)
(c) Change in internal energy of the gas:
Since ∆U = Q - W, we can substitute the values we have:
∆U = 1.430 kJ - W
∆U = 1.430 kJ + (0.966 atm)(∆V)
(d) Final volume of the gas:
The final volume (Vf) can be found by rearranging the ideal gas law equation:
Vf = nRT/P
Vf = (1.20 mol)(0.0821 L·atm/(mol·K))(Tf)/(0.966 atm)
Now we have the equations to calculate the final temperature (Tf), work done by the gas (W), change in internal energy (∆U), and final volume of the gas (Vf).
To calculate these values, we can use the ideal gas law and the first law of thermodynamics.
First, let's write down the known information:
Initial conditions:
- Amount of gas (n): 1.20 mol
- Initial temperature (T): 25°C = 298 K
- Initial pressure (P): 0.966 atm
- Initial volume (V): unknown
Heat added to the gas (q): 1.430 kJ
We will assume helium gas follows ideal gas behavior.
a. Final temperature of gas:
To find the final temperature (Tf), we can use the equation:
(P * V) / T = n * R
Where:
- P is the pressure of the gas
- V is the volume of the gas
- T is the temperature of the gas
- n is the number of moles of gas
- R is the ideal gas constant (0.0821 L * atm / (mol * K))
Since the pressure is constant (ambient pressure), we can rewrite the equation as:
V / T = n * R / P
We can rearrange this equation to solve for the final temperature (Tf):
Tf = (V * n * R) / (P)
We know the values of n, R, P, but we still need to determine the final volume (Vf) to calculate the final temperature.
b. Work done by the gas:
The work done by the gas (W) can be calculated by the equation:
W = P * ΔV
Where:
- W is the work done by the gas
- P is the constant pressure of the gas
- ΔV is the change in volume of the gas
We need to determine the change in volume (ΔV) to calculate the work done by the gas.
c. Change in internal energy of the gas:
The change in internal energy (ΔU) of the gas can be calculated using the first law of thermodynamics:
ΔU = q - W
Where:
- ΔU is the change in internal energy of the gas
- q is the heat added to the gas
- W is the work done by the gas
We already know the value of q, and we can calculate the value of W (using the volume change, which we will determine), to find ΔU.
d. Final volume of the gas:
We still need to determine the final volume (Vf) to calculate the other values. To find Vf, we can use the ideal gas law:
(P * V) / T = n * R
Where:
- P is the pressure of the gas
- V is the volume of the gas
- T is the temperature of the gas
- n is the number of moles of gas
- R is the ideal gas constant (0.0821 L * atm / (mol * K))
We can rearrange this equation to solve for the final volume (Vf):
Vf = (P * V * Tf) / (P * T)
Since we don't have the initial volume (Vi), we need to find it.
To find the initial volume, we can use the ideal gas law at the initial temperature and pressure (T and P):
(P * Vi) / T = n * R
Rearranging the equation to solve for Vi:
Vi = (P * n * R) / T
We can substitute this value of Vi into the equation for Vf to calculate the final volume.