Question

Over limited intervals of temperature, the saturation pressure-temperature curve for two-phase liquid-vapor states can be represented by an equation of the form $\ln p_{\text{sat }}=A-B/T$, where $A$ and $B$ are constants. Derive the following expression relating any three states on such a portion of the curve: $$\frac{p_{\text{sat, }3}}{p_{\text{sat, }1}}={\left(\frac{p_{\text{sat, }2}}{p_{\text{sat },1}}\right)}^{\tau }$$ where $\tau =T_2(T_3-T_1)/T_3(T_2-T_1)$.

Short answer

$$\frac{p_{sat,3}}{p_{sat,1}}={\left(\frac{p_{sat,2}}{p_{sat,1}}\right)}^{\tau }$$

Step-by-step solution

Write down the given equation

The given equation is $$ln(p_{sat})=A-\frac{B}{T}$$

Express the equation for three different states

Write the equation for three different states $$\ln (p_{sat,1})=A-\frac{B}{T_1}$$ $$\ln (p_{sat,2})=A-\frac{B}{T_2}$$ $$\ln (p_{sat,3})=A-\frac{B}{T_3}$$

Isolate the natural logarithms

Isolate the natural logarithms of the pressure values: $$\ln (p_{sat,1})=A-\frac{B}{T_1}\Rightarrow A=\ln (p_{sat,1})+\frac{B}{T_1}$$ $$\ln (p_{sat,2})=A-\frac{B}{T_2}\Rightarrow A=\ln (p_{sat,2})+\frac{B}{T_2}$$ $$\ln (p_{sat,3})=A-\frac{B}{T_3}\Rightarrow A=\ln (p_{sat,3})+\frac{B}{T_3}$$

Equate the expressions

Since all expressions are equal to A, we equate the expressions:$$\ln (p_{sat,1})+\frac{B}{T_1}=\ln (p_{sat,2})+\frac{B}{T_2}\Rightarrow \ln (p_{sat,2})-\ln (p_{sat,1})=\frac{B}{T_1}-\frac{B}{T_2}$$ $$\ln (p_{sat,2})+\frac{B}{T_2}=\ln (p_{sat,3})+\frac{B}{T_3}\Rightarrow ln(p_{sat,3})-\ln (p_{sat,2})=\frac{B}{T_2}-\frac{B}{T_3}$$

Combine the equations

Combine both expressions to form the relationship:$$\frac{\ln (p_{sat,3})-\ln (p_{sat,2})}{\ln (p_{sat,2})-\ln (p_{sat,1})}=\frac{\frac{B}{T_2}-\frac{B}{T_3}}{\frac{B}{T_1}-\frac{B}{T_2}}$$

Simplify the expressions

Simplify both sides of the equation: $$\ln \left(\frac{p_{sat,3}}{p_{sat,2}}\right)/\ln \left(\frac{p_{sat,2}}{p_{sat,1}}\right)=\frac{T_2(T_1-T_3)}{T_3(T_1-T_2)}$$

Express in terms of $\tau$

Recognize that: $$\tau =\frac{T_2(T_3-T_1)}{T_3(T_2-T_1)}$$ Rearranging gives: $$\frac{T_2(T_1-T_3)}{T_3(T_1-T_2)}=-\tau \ln \left(\frac{p_{sat,3}}{p_{sat,2}}\right)=-\tau \ln \left(\frac{p_{sat,2}}{p_{sat,1}}\right)$$ Exponentiating both sides yields: $$\frac{p_{sat,3}}{p_{sat,2}}={\left(\frac{p_{sat,2}}{p_{sat,1}}\right)}^{-\tau }$$

Final expression

Rewriting the expression: $$\frac{p_{sat,3}}{p_{sat,1}}={\left(\frac{p_{sat,2}}{p_{sat,1}}\right)}^{\tau }$$

Key concepts

Thermodynamics in Saturation Pressure-Temperature Relationships

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. It plays a crucial role in understanding how substances change from one phase to another. In this exercise, we are looking at the saturation pressure-temperature relationship for a liquid-vapor system.
This relationship can be modeled by: $\ln (p_{\text{sat}})=A-\frac{B}{T}$ where $p_{\text{sat}}$ represents the saturation pressure, $T$ is the temperature, and $A$ and $B$ are constants specific to the substance.
Thermodynamics tells us that there is a precise temperature at which a substance can exist in equilibrium between liquid and vapor phases, and this is critical for predicting and calculating various properties of materials.

Phase Equilibrium and Multi-State Relationships

Phase Equilibrium refers to the state where a substance can exist in two phases at the same time without changing its state entirely. This is essential for understanding the saturation pressure-temperature relationship explained earlier.
In the original exercise, we derived a relationship connecting the saturation pressures at three different temperatures. This equilibrium condition ensures that we can find a consistent relationship between pressure and temperature using thermodynamic principles.
When we consider three states: State 1, State 2, and State 3, each represented by $p_{\text{sat,1}}$, $p_{\text{sat,2}}$, and $p_{\text{sat,3}}$ respectively, we can connect these points through $\frac{p_{\text{sat,3}}}{p_{\text{sat,1}}}={\left(\frac{p_{\text{sat,2}}}{p_{\text{sat,1}}}\right)}^{\tau }$.
$\tau$ is defined as $\tau =\frac{T_2(T_3-T_1)}{T_3(T_2-T_1)}$. This equation allows us to predict the saturation pressure at different states if we know the pressure at other states, showcasing the interconnectedness of phase equilibrium.

Importance of Natural Logarithm in Thermodynamics

Natural Logarithm is a mathematical function that is particularly useful in thermodynamics and various other fields of science and engineering. It is denoted as $\ln (x)$ and has unique properties that simplify complex multiplicative processes into additive ones.
In the context of the saturation pressure-temperature relationship, the natural logarithm helps linearize the exponential relationship between pressure and temperature. This provides a clearer and more straightforward means for analysis and derivation.
By taking the natural logarithm of the saturation pressure equation, $ln(p_{\text{sat}})=A-\frac{B}{T}$, the complex relationship becomes a linear one, making it easier to solve for unknown values and derive further relationships. This natural logarithm-based expression is key to deriving further expressions, such as the relation involving three states. As we manipulate these logarithmic equations, we can eventually exponentiate in the final steps to find the required expressions in their simplified form.