Problem 156 - Calculus 3

Question

The function \(P(T, V)=\frac{n R T}{V}\) gives the pressure at a point in a gas as a function of temperature \(T\) and volume V. The letters \(n\) and \(R\) are constants. Find \(\frac{\partial P}{\partial V}\) and \(\frac{\partial P}{\partial T}\) and explain what these quantities represent.

Step-by-Step Solution

Answer

\(\frac{\partial P}{\partial V} = -\frac{nRT}{V^2}\), pressure decreases with volume. \(\frac{\partial P}{\partial T} = \frac{nR}{V}\), pressure increases with temperature.

Step 1: Identify the Function

The given function is \( P(T, V) = \frac{nRT}{V} \). This equation represents the pressure of a gas in terms of the temperature \( T \) and volume \( V \).

Step 2: Partial Derivative with Respect to Volume

To find \( \frac{\partial P}{\partial V} \), we differentiate \( P(T, V) = \frac{nRT}{V} \) with respect to \( V \), treating \( T \) as a constant. Using the power rule, \( \frac{d}{dV} \left( V^{-1} \right) = -V^{-2} \), we have:\[\frac{\partial P}{\partial V} = \frac{nRT}{V^2} (-1) = -\frac{nRT}{V^2}.\]

Step 3: Interpretation: \(\frac{\partial P}{\partial V}\)

The term \( \frac{\partial P}{\partial V} = -\frac{nRT}{V^2} \) represents the rate of change of pressure with respect to changes in volume, holding temperature constant. Since the expression is negative, it indicates that pressure decreases as volume increases (assuming other factors remain constant).

Step 4: Partial Derivative with Respect to Temperature

To find \( \frac{\partial P}{\partial T} \), we differentiate \( P(T, V) = \frac{nRT}{V} \) with respect to \( T \), treating \( V \) as a constant. Since \( nR \) and \( V \) are constants, the derivative simplifies to:\[\frac{\partial P}{\partial T} = \frac{nR}{V}.\]

Step 5: Interpretation: \(\frac{\partial P}{\partial T}\)

The term \( \frac{\partial P}{\partial T} = \frac{nR}{V} \) represents the rate of change of pressure with respect to changes in temperature, holding volume constant. It shows that pressure increases as temperature increases (assuming volume remains constant).

Key Concepts

Pressure Equation

The pressure equation for a gas is a fundamental concept in thermodynamics, especially when dealing with ideal gases. The equation we're focusing on is given by the function \( P(T, V) = \frac{nRT}{V} \). This equation allows us to see how the pressure \( P \) changes when the temperature \( T \) and volume \( V \) change, assuming they both contribute to how gases behave under various conditions.

Here, \( n \) denotes the number of moles of the gas and \( R \) is the universal gas constant. Both \( n \) and \( R \) are constants, meaning their values do not change as \( T \) and \( V \) change.

The equation shows that:

• If the temperature \( T \) increases while volume \( V \) stays the same, the pressure \( P \) will increase.
• If the volume \( V \) increases while temperature \( T \) remains constant, the pressure \( P \) will decrease.

This equation is a great example of how mathematical equations help us understand physical laws and phenomena.

Rate of Change

The rate of change in mathematics tells us how one variable changes in relation to another. For our gas pressure function, we look at how pressure changes when volume or temperature changes. This involves calculating partial derivatives.

When we take the partial derivative of pressure with respect to volume \( \frac{\partial P}{\partial V} \), it shows the rate at which the pressure changes if only the volume changes, assuming temperature remains constant. For the given function:

• The derivative is \( -\frac{nRT}{V^2} \), which signifies that pressure decreases if volume increases, keeping temperature constant.

Similarly, when we take the partial derivative of pressure with respect to temperature \( \frac{\partial P}{\partial T} \), it tells us how pressure changes when only temperature changes, leaving volume constant. For the function, it becomes:

• \( \frac{nR}{V} \), showing that pressure increases as temperature goes up, with a constant volume.

In essence, these derivatives reflect how sensitive the pressure is to small changes in temperature or volume.

Temperature and Volume

Temperature and volume are two crucial factors that affect the behavior of gases, covered under the broader Umbrella of the ideal gas law. Here's how they play their roles:Temperature:

• Temperature is a measure of the average kinetic energy of the gas molecules. Higher temperatures mean these molecules move more vigorously, leading to increased pressure.
• According to our pressure function, an increase in temperature \( T \) results in a proportional increase in pressure, provided the volume \( V \) remains fixed.

Volume:

• The volume of a gas dictates how much space the gas particles have to move around. More volume gives them more space to spread, which decreases the collisions against the container walls, thus, reducing pressure.
• In our function, as volume \( V \) increases, the pressure \( P \) decreases if the temperature \( T \) is kept constant.

Understanding these relationships helps predict how changes in temperature or volume will affect the pressure, a key concept in fields like chemistry and physics.