Question

Liquid/vapor saturation pressure Psat Psat is often represented as a function of temperature by the Antoine equation, which can be written in the form: log⁡10Psat/(torr)=a−bt/∘C+clog10​Psat/(torr)=a−t/∘C+cb​ Here, parameters a, b, and c are substance-specific constants. Suppose this equation is to be rewritten in the equivalent form: ln⁡Psat⁡/kPa=A−BT/K+ClnPsat/kPa=A−T/K+CB​ Show how the parameters in the two equations are related.

Answer 1

The relationship between the parameters in the Antoine equation for vapor pressure, log⁡(Psat/torr) = a - bt/°C + c, and the equivalent form ln(Psat/kPa) = A - BT/K + C, is given by: A = a - log(7.50062), B = b/2.303, and C = c.

Step-by-step explanation:

The relationship between log⁡(Psat/torr) and ln(Psat/kPa) can be established using the conversion factor between torr and kPa, which is 7.50062 torr = 1 kPa. To derive A, consider the conversion of units: ln(Psat/kPa) = ln(Psat/7.50062 torr) + ln(7.50062 torr/kPa). This simplifies to ln(Psat/kPa) = log⁡(Psat/torr) - log(7.50062). Hence, A = a - log(7.50062) as log⁡(Psat/torr) = a - bt/°C + c becomes ln(Psat/kPa) = A - BT/K + C.

To relate the temperature coefficients, note that the relationship between °C and K is given by K = °C + 273.15. Thus, B = b/2.303 to convert the temperature coefficient from the units of °C to K.

Finally, parameter C remains unchanged as it represents the substance-specific constant in both forms of the equation. Therefore, the relationship between the parameters in the two equations for vapor pressure involves adjusting A to account for the unit conversion from torr to kPa, dividing b by 2.303 for the temperature coefficient, and maintaining C as the substance-specific constant.

Answer 2

To convert parameters from the Antoine equation to the equivalent natural logarithm equation, conversion from log base-10 to natural logarithm is applied to the coefficients. The pressure from torr to kPa and temperature from Celsius to Kelvin are also accounted for, leading to new coefficients A, B, and a modified C.

Step-by-step explanation:

To relate the Antoine equation parameters for liquid/vapor saturation pressure Psat given as log₁₀Psat/(torr) = a - bt/°C + c to the parameters for the equivalent form ln Psat/kPa = A - BT/K + C, a conversion process is required due to the difference in the logarithm bases and the units of pressure and temperature.

First, we need to convert the base-10 logarithm to a natural logarithm. The conversion can be done using the relation log₁₀(x) = ln(x)/ln(10). Thus, the Antoine equation becomes ln Psat/(torr) = (a ln(10)) - (b ln(10))t/°C + (c ln(10)).

Then, we have to convert the pressure from torr to kPa and the temperature from Celsius to Kelvin. Since 1 torr is approximately 0.133322 kPa and the conversion from degrees Celsius to Kelvin is T(K) = t(°C) + 273.15, the coefficients will be modified accordingly.

The final form ln Psat/kPa = A - BT/K + C will have the parameters related to the original Antoine coefficients (a, b, c) as follows:

• A = a ln(10) converts the constant for base-10 to natural logarithm and remains unaffected by unit changes
• B = b ln(10) similarly converts the coefficient associated with temperature and is unaltered by unit changes
• C must be adjusted for the pressure unit conversion factor between torr and kPa