By using the Clausius-Clapeyron equation,lnP=Hvap/RT +C , derive the relationship between the vapor pressures,P1 and P2, and the absolute temperatures at which they were measured,T1 and T2 .
ln(P1/P2)={-ÄHvap[(1/T1)-(1/T2)]}/R
To derive the relationship between vapor pressures P1 and P2 and the absolute temperatures T1 and T2 using the Clausius-Clapeyron equation, lnP = Hvap/RT + C, we can start by considering the ratio of the vapor pressures:
ln(P2/P1) = Hvap/R * (1/T1 - 1/T2)
In this equation, Hvap is the enthalpy of vaporization, R is the ideal gas constant, T1 and T2 are the absolute temperatures at which the vapor pressures P1 and P2 were measured.
Now, to simplify this equation, we can take exponential of both sides:
e^(ln(P2/P1)) = e^(Hvap/R * (1/T1 - 1/T2))
P2/P1 = e^(Hvap/R * (1/T1 - 1/T2))
Now, we can rearrange this equation to obtain the relationship between the vapor pressures P1 and P2 and the absolute temperatures T1 and T2:
P2 = P1 * e^(Hvap/R * (1/T1 - 1/T2))
Therefore, the relationship between the vapor pressures P1 and P2 and the absolute temperatures T1 and T2 is given by the equation:
P2 = P1 * e^(Hvap/R * (1/T1 - 1/T2))
To derive the relationship between the vapor pressures (P1 and P2) and the absolute temperatures (T1 and T2), we can start with the Clausius-Clapeyron equation:
ln(P) = Hvap / RT + C
where:
P is the vapor pressure of the substance,
Hvap is the enthalpy of vaporization of the substance,
R is the ideal gas constant,
T is the absolute temperature, and
C is a constant
To find the relationship between two different vapor pressures (P1 and P2) at their respective absolute temperatures (T1 and T2), we need to compare the two equations for ln(P).
For vapor pressure P1 at temperature T1:
ln(P1) = Hvap / (R * T1) + C ----(equation 1)
For vapor pressure P2 at temperature T2:
ln(P2) = Hvap / (R * T2) + C ----(equation 2)
To derive the relationship, we can first subtract equation 2 from equation 1:
ln(P1) - ln(P2) = (Hvap / (R * T1) + C) - (Hvap / (R * T2) + C)
Now, we can simplify the equation further:
ln(P1) - ln(P2) = (Hvap / (R * T1)) - (Hvap / (R * T2))
Using logarithm properties, we know that ln(P1) - ln(P2) is equal to the natural logarithm of the ratio of P1 to P2:
ln(P1) - ln(P2) = ln(P1/P2)
Thus, the equation becomes:
ln(P1/P2) = (Hvap / (R * T1)) - (Hvap / (R * T2))
Now, we can rearrange the equation to solve for the relationship between the vapor pressures and temperatures:
ln(P1 / P2) = Hvap / R * (1/T1 - 1/T2)
Finally, we can exponentiate both sides of the equation to remove the natural logarithm:
P1 / P2 = e^(Hvap / R * (1/T1 - 1/T2))
So, the derived relationship between the vapor pressures (P1 and P2) and the absolute temperatures (T1 and T2) can be described by:
P1 / P2 = e^(Hvap / R * (1/T1 - 1/T2))
This equation demonstrates how the ratio of vapor pressures depends on the difference in the reciprocal of absolute temperatures, scaled by the enthalpy of vaporization and the ideal gas constant.