A sample of monatomic gas (ã = 5/3) is expanded adiabatically from an initial temperature, volume, and pressure of 230. °C, 7.00 litres, and 7.00 kPa, respectively, to a final volume of 10.0 litres.
(a) What is the final pressure of the gas? (answer 3.86 kPa)
(b) How much work is done by the gas during the adiabatic expansion? (answer 15.6 J)
To solve this problem, we will use the adiabatic equation for an ideal gas:
P1 * V1^γ = P2 * V2^γ
Where:
P1 and P2 are the initial and final pressures,
V1 and V2 are the initial and final volumes, and
γ is the heat capacity ratio, which is equal to 5/3 for a monatomic gas.
(a) To find the final pressure, we can rearrange the equation as follows:
P2 = (P1 * V1^γ) / V2^γ
Substitute the given values into the equation:
P2 = (7.00 kPa * (230 °C + 273.15 K)^5/3) / (10.0 L)^5/3
Now, we need to convert the initial temperature from Celsius to Kelvin:
T1 = 230 °C + 273.15 K = 503.15 K
Substitute the values into the equation:
P2 = (7.00 kPa * (503.15 K)^5/3) / (10.0 L)^5/3
Calculate the values inside the parentheses:
P2 = (7.00 kPa * 1195204.3 K^(5/3)) / (10.0 L)^5/3
Now, calculate the final pressure:
P2 ≈ 3.86 kPa
So, the final pressure of the gas (answer to part a) is approximately 3.86 kPa.
(b) To find the work done by the gas during the adiabatic expansion, we can use the formula:
W = (P1 * V1 - P2 * V2) / (γ - 1)
Substitute the given values into the equation:
W = ((7.00 kPa * 7.00 L) - (3.86 kPa * 10.0 L)) / (5/3 - 1)
Calculate the numerator:
W = (49 kPa·L - 38.6 kPa·L) / (2/3)
W = 10.4 kPa·L / (2/3)
Multiply by the reciprocal to divide by a fraction:
W = 10.4 kPa·L * (3/2)
W = 15.6 kPa·L
Finally, convert kPa·L to joules (J) since 1 kPa·L = 1 J:
W ≈ 15.6 J
So, the work done by the gas during the adiabatic expansion (answer to part b) is approximately 15.6 J.