An ideal monatomic gas initially has a temperature of 338 K and a pressure of 6.96 atm. It is to expand from volume 435 cm3 to volume 1310 cm3. If the expansion is isothermal, what are (a) the final pressure (in atm) and (b) the work done by the gas? If, instead, the expansion is adiabatic, what are (c) the final pressure (in atm) and (d) the work done by the gas?
To solve these questions, we can use the ideal gas law and the equations for isothermal and adiabatic processes.
First, let's calculate the final pressure (P2) and the work done (W) for the isothermal process.
(a) To find the final pressure (P2), we can use the equation for an isothermal process:
P1 * V1 = P2 * V2,
where P1 and V1 are the initial pressure and volume, and V2 is the final volume.
Given:
P1 = 6.96 atm
V1 = 435 cm^3
V2 = 1310 cm^3
Plug in these values into the equation:
6.96 atm * 435 cm^3 = P2 * 1310 cm^3.
Now, solve for P2:
P2 = (6.96 atm * 435 cm^3) / 1310 cm^3.
Calculating this expression gives us the final pressure (P2).
(b) To find the work done by the gas, we can use the equation for work in an isothermal process:
W = nRT * ln(V2/V1),
where n is the number of moles of the gas, R is the ideal gas constant, T is the temperature in Kelvin, and ln(V2/V1) is the natural logarithm of the ratio of final and initial volumes.
Given:
T = 338 K (initial temperature)
V1 = 435 cm^3
V2 = 1310 cm^3
Plug in these values into the equation:
W = nRT * ln(1310 cm^3 / 435 cm^3).
Now, solve this expression to find the work done by the gas (W) in the isothermal process.
Now, let's move on to the adiabatic process:
(c) To find the final pressure (P2) in the adiabatic process, we can use the adiabatic equation:
P1 * V1^γ = P2 * V2^γ,
where γ is the heat capacity ratio, which depends on the specific gas. For a monatomic gas, γ is equal to 5/3.
Given:
P1 = 6.96 atm
V1 = 435 cm^3
V2 = 1310 cm^3
γ = 5/3
Plug in these values into the equation:
6.96 atm * (435 cm^3)^5/3 = P2 * (1310 cm^3)^5/3.
Now, solve for P2:
P2 = (6.96 atm * (435 cm^3)^5/3) / (1310 cm^3)^5/3.
Calculating this expression gives us the final pressure (P2) in the adiabatic process.
(d) To find the work done by the gas in the adiabatic process, we can use the equation:
W = (P2 * V2 - P1 * V1) / (γ - 1).
Plug in the values of P2, V2, P1, V1, and γ into this equation and solve to find the work done (W) by the gas in the adiabatic process.
Remember to convert the volumes from cm^3 to a consistent unit, such as m^3, if necessary, and ensure that all calculations are done using consistent units.