An ideal gas is allowed to expand isothermally
from 2.00 L at 5.00 atm in two steps:
a) against a constant external pressure of
3.00 atm, followed by
b) against a constant external pressure of
2.00 atm.
Calculate q and w. (101.33 J = 1 L atm)
To calculate the heat transfer (q) and work done (w) during an isothermal expansion, we can use the formulas:
q = -nRT ln(V2/V1)
w = -q
where:
- q is the heat transfer
- n is the number of moles of gas
- R is the ideal gas constant
- T is the temperature in Kelvin
- V1 is the initial volume of the gas
- V2 is the final volume of the gas
First, we need to find the number of moles of gas. The ideal gas law can help with this:
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
- T is the temperature in Kelvin
Let's solve for n using the initial conditions:
n = PV / RT
= (5.00 atm * 2.00 L) / (0.0821 L atm/mol K * T)
Now let's calculate q:
q = -nRT ln(V2/V1)
= -[(5.00 atm * 2.00 L) / (0.0821 L atm/mol K * T)] * (0.0821 L atm/mol K * T) * ln(V2/2.00 L)
= - (5.00 atm * 2.00 L) * ln(V2/2.00 L)
Similarly, we can calculate w:
w = -q
Now let's substitute the given external pressures for V2 and perform the calculations.
For part a:
V2 = 3.00 atm
q_a = - (5.00 atm * 2.00 L) * ln(3.00 atm/2.00 L)
w_a = -q_a
For part b:
V2 = 2.00 atm
q_b = - (5.00 atm * 2.00 L) * ln(2.00 atm/2.00 L)
w_b = -q_b
By substituting the values of V2 into the equations, you can calculate q_a, w_a, q_b, and w_b.