Calculate DS if the temperature of 3 moles of an ideal gas with Cv,m = 1.5R is
increased from 200 K to 500 K under conditions of (a) constant pressure and (b)
constant volume.
To calculate the change in entropy (ΔS) of an ideal gas, we can use the equation:
ΔS = n * Cv * ln(T2 / T1)
where:
ΔS is the change in entropy
n is the number of moles of the gas
Cv is the molar specific heat capacity of the gas at constant volume
T1 is the initial temperature in Kelvin
T2 is the final temperature in Kelvin
ln(x) is the natural logarithm of x
In this case, we are given:
n = 3 moles
Cv,m = 1.5R (where R is the molar gas constant)
T1 = 200 K
T2 = 500 K
For part (a) where the conditions are constant pressure, we need to use Cv,m – R = Cp,m
So, Cp,m = 1.5R + R = 2.5R
Therefore, for part (a), we can calculate ΔS using the equation:
ΔS = n * Cp * ln(T2 / T1)
Now substituting the values:
ΔS(a) = 3 * 2.5R * ln(500 / 200)
For part (b) where the conditions are constant volume, we can use Cv,m directly.
Therefore, for part (b), we can calculate ΔS using the equation:
ΔS(b) = n * Cv * ln(T2 / T1)
Substituting the values:
ΔS(b) = 3 * 1.5R * ln(500 / 200)
Now we have the equations to calculate the change in entropy for both cases. We can substitute the value of R (the molar gas constant) to get the final numerical answer.