Problem 68 - Algebra and Trigonometry with Analytic Geometry

Question

A liquid's vapor pressure \(P\) (in \(\mathrm{lb} / \mathrm{in}^{2}\) ), a measure of its volatility, is related to its temperature \(T\) (in \({ }^{\circ} \mathrm{F}\) ) by the Antoine equation $$ \log P=a+\frac{b}{c+T} $$ where \(a, b\), and \(c\) are constants. Vapor pressure increases rapidly with an increase in temperature. Express \(P\) as a function of \(T\).

Step-by-Step Solution

Answer

\( P = 10^{(a + \frac{b}{c+T})} \) is the function of \( T \).

Step 1: Understand the Antoine Equation

The given equation is \( \log P = a + \frac{b}{c+T} \). This equation relates the logarithm of the vapor pressure \( P \) to the temperature \( T \) using constants \( a \), \( b \), and \( c \). To express \( P \) as a direct function of \( T \), we need to solve for \( P \).

Step 2: Isolate the Logarithmic Expression

The equation \( \log P = a + \frac{b}{c+T} \) is already isolated in terms of \( \log P \). Our task is to derive \( P \), not \( \log P \), so we need to eliminate the logarithm.

Step 3: Exponentiate Both Sides

To remove the logarithm, exponentiate both sides of the equation using base 10: \[ P = 10^{(a + \frac{b}{c+T})} \]This step converts the equation into a form that directly expresses \( P \) as a function of \( T \), by applying the exponential function to both sides.

Step 4: Simplify the Expression for \( P \)

After exponentiating, the expression \( P = 10^{(a + \frac{b}{c+T})} \) clearly shows that \( P \) depends on the constants \( a \), \( b \), and \( c \), and is a function of the temperature \( T \). Many real-world applications require this direct relationship as it allows for the computation of vapor pressure at a given temperature.

Key Concepts

Vapor Pressure

Vapor pressure is an important concept in understanding the behavior of liquids. It refers to the pressure exerted by a vapor in equilibrium with its liquid state. This equilibrium is crucial as it shows the point where the rate of evaporation equals the rate of condensation. Vapor pressure gives insights into how volatile a liquid is; a higher vapor pressure at a given temperature indicates a more volatile liquid.

Key points to remember about vapor pressure:

• It is affected by temperature; as temperature increases, vapor pressure also increases.
• A liquid with high vapor pressure evaporates more quickly than one with low vapor pressure.
• The Antoine equation is commonly used to calculate the vapor pressure of a substance at different temperatures.

Vapor pressure is vital in fields such as chemistry and meteorology, influencing how substances interact and react.

Temperature Dependence

Temperature plays a pivotal role in determining the vapor pressure of a liquid. Essentially, temperature provides the energy required for molecules in the liquid to overcome intermolecular forces and enter the gaseous state.

The Antoine equation highlights this temperature dependence through its mathematical structure:

• Increased temperature results in increased kinetic energy of molecules.
• The fraction \( \frac{b}{c+T} \) in the Antoine equation shows how changes in temperature can affect vapor pressure.
• In the equation's setup, as temperature \( T \) rises, the denominator \( c+T \) becomes larger, reducing the fraction, consequently reducing the logarithmic factor, \( \frac{b}{c+T} \), resulting in an increase in \( P \).

The strong relationship between temperature and vapor pressure is why these two variables are often studied together in thermodynamics and physical chemistry.

Exponential Functions

Exponential functions are a cornerstone of the Antoine equation's structure. With the form \( P = 10^{(a + \frac{b}{c+T})} \), vapor pressure is represented as a function of temperature, an exponential function highlights this dependency.

An exponential function implies:

• A rapid increase or decrease of a quantity with respect to changes in other variables, like temperature.
• In our context, it means that even small changes in temperature can lead to significant changes in vapor pressure.
• The base of the exponent, which is 10 in the Antoine equation, defines the rapidity of these changes.

Understanding exponential functions is essential because they provide insights into how sensitive a process or reaction is to changes, a vital consideration in both experimental and real-world applications.