One mole of an ideal gas does 3400 J of work on its surroundings as it expands isothermally to a final pressure of 1.00 atm and volume of 22.0 L.
(a) Determine the initial volume in m^3.
i tried using the formula w=integral f/i pdv
3400J= -(1.00atm)(.022m^3Vf-Vi)
i eneded with 1.55 x10^5 but htis is not the right answer. HELP PLS!
Isothermal expansion
W = integral p dv
p = n R T/v, T constant so
W = n R T integral dv/v
W = n R T ln v from v1 to v2
W = n R T ln(v2/v1)
here
n = 1
R = R =.08206 L atm/mol degK or 8.31 J/mol deg K
Need T, use v2
T = p2 v2/nR = 1*22/1*.082
T = 268 deg K
now we want to use Joules and scientific notaton R
V2 = 22 L = 22* 10^-3 m^3
3400 = 1 * 8.31 * 268 ln (22*10^-3/v)
ln(.022/v)= 1.53
.022/v = 4.6
v = .00478 m^3 = 4.78 Liters
To find the initial volume in m^3, you can start by rearranging the equation you used. The equation you used is:
w = -∫PdV
Since the process is isothermal, the ideal gas equation can be written as:
PV = nRT
Where:
P = pressure (in atm)
V = volume (in liters)
n = number of moles
R = ideal gas constant (0.0821 atm·L/mol·K for this problem)
T = temperature (in Kelvin)
Since the gas is ideal and the process is isothermal, the temperature remains constant, so we can write:
PV = constant
Now, let's find the value of the constant using the initial conditions.
Given:
w = 3400 J
Pf = 1.00 atm
Vf = 22.0 L
Using the ideal gas equation, we can write:
Vi * Pi = Vf * Pf
Substituting the values:
Vi * Pi = 22.0 L * 1.00 atm
Now, we need to convert the pressure to Pa and the volume to m^3 since the unit of work is J and not atm·L:
1 atm = 101325 Pa
1 L = 0.001 m^3
So, we have:
Vi * (Pi * 101325 Pa/atm) = (22.0 L * 0.001 m^3/L) * 101325 Pa/atm
Simplifying:
Vi * (Pi * 101325) = 22.0 * 0.101325
Vi * Pi = 2.22715
Now, let's substitute this value of Vi * Pi into the work equation:
3400 J = -(Vi * Pi) * (Vf - Vi)
Rearranging the equation:
Vi * Pi = -3400 J / (Vf - Vi)
Substituting the values:
2.22715 = -3400 J / (22.0 L - Vi)
Now, solve for Vi:
Vi = (3400 J / 2.22715) - 22.0 L
Vi ≈ 1526.04 L
Finally, convert to m^3:
Vi = 1526.04 L * 0.001 m^3/L
Vi ≈ 1.53 m^3