Problem 156
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
Verified Answer
The partial derivatives are \( \frac{\partial P}{\partial V} = -\frac{nRT}{V^2} \) and \( \frac{\partial P}{\partial T} = \frac{nR}{V} \).
1Step 1: Understand the Function
The given function is the ideal gas law expressed as \( P(T, V) = \frac{nRT}{V} \), where \( P \) is the pressure, \( T \) is the temperature, \( V \) is the volume, \( n \) is the number of moles of gas, and \( R \) is the gas constant.
2Step 2: Partial Derivative with respect to Volume \( V \)
To find \( \frac{\partial P}{\partial V} \), we differentiate \( P(T, V) = \frac{nRT}{V} \) with respect to \( V \), treating \( T \) as a constant.\[ \frac{\partial P}{\partial V} = \frac{\partial}{\partial V} \left( \frac{nRT}{V} \right) = -\frac{nRT}{V^2} \]
3Step 3: Partial Derivative with respect to Temperature \( T \)
To find \( \frac{\partial P}{\partial T} \), we differentiate \( P(T, V) = \frac{nRT}{V} \) with respect to \( T \), treating \( V \) as a constant.\[ \frac{\partial P}{\partial T} = \frac{\partial}{\partial T} \left( \frac{nRT}{V} \right) = \frac{nR}{V} \]
4Step 4: Interpret the Derivatives
The derivative \( \frac{\partial P}{\partial V} = -\frac{nRT}{V^2} \) indicates how the pressure \( P \) changes with a change in volume \( V \), holding temperature \( T \) constant. It shows that an increase in volume decreases the pressure, as expected from Boyle's law.The derivative \( \frac{\partial P}{\partial T} = \frac{nR}{V} \) indicates how the pressure \( P \) changes with a change in temperature \( T \), holding volume \( V \) constant. It shows that increasing the temperature increases the pressure, consistent with Charles's law.
Key Concepts
Ideal Gas LawBoyle's LawCharles's LawMultivariable Calculus
Ideal Gas Law
The ideal gas law is a fundamental equation in thermodynamics and chemistry. It describes the behavior of an ideal gas by relating pressure (\(P\)), volume (\(V\)), temperature (\(T\)), and the amount of gas (\(n\)). The equation is represented as:
\[ P(T, V) = \frac{nRT}{V} \]
Here, \(R\) is known as the universal gas constant. The ideal gas law assumes a perfect and theoretical gas where molecules do not interact and occupy no volume themselves. Practically, it provides a useful approximation for real gases under many conditions. Its applications are widespread, from predicting weather patterns to determining the behavior of gases in chemical reactions.
\[ P(T, V) = \frac{nRT}{V} \]
Here, \(R\) is known as the universal gas constant. The ideal gas law assumes a perfect and theoretical gas where molecules do not interact and occupy no volume themselves. Practically, it provides a useful approximation for real gases under many conditions. Its applications are widespread, from predicting weather patterns to determining the behavior of gases in chemical reactions.
Boyle's Law
Boyle's Law is a specific principle derived from the ideal gas law. It describes how the pressure of a gas tends to increase as the volume of the container decreases, if the temperature remains constant. Mathematically, it is expressed as:
\[ P \times V = \, \text{constant \(k\)} \]
When examining the partial derivative \( \frac{\partial P}{\partial V} \), you observe how pressure changes with volume changes while temperature is constant. The derivative \( -\frac{nRT}{V^2} \) supports Boyle's Law, indicating an inverse relationship between pressure and volume. This means that pressure decreases if volume increases and vice versa.
\[ P \times V = \, \text{constant \(k\)} \]
When examining the partial derivative \( \frac{\partial P}{\partial V} \), you observe how pressure changes with volume changes while temperature is constant. The derivative \( -\frac{nRT}{V^2} \) supports Boyle's Law, indicating an inverse relationship between pressure and volume. This means that pressure decreases if volume increases and vice versa.
Charles's Law
Charles's Law explains the direct relationship between temperature and volume at constant pressure. It states that the volume of a gas is directly proportional to its temperature when pressure is kept constant:
\[ \frac{V}{T} = \, \text{constant \(c\)} \]
In terms of partial derivatives, examining \( \frac{\partial P}{\partial T} \) reveals how pressure changes with a change in temperature, assuming volume is constant. The derivative \( \frac{nR}{V} \), derived from the ideal gas law, illustrates Charles's Law by showing pressure increases with temperature. This relationship helps in understanding how heating a gas at constant volume increases its pressure, which is essential in designing pressure vessels and engines.
\[ \frac{V}{T} = \, \text{constant \(c\)} \]
In terms of partial derivatives, examining \( \frac{\partial P}{\partial T} \) reveals how pressure changes with a change in temperature, assuming volume is constant. The derivative \( \frac{nR}{V} \), derived from the ideal gas law, illustrates Charles's Law by showing pressure increases with temperature. This relationship helps in understanding how heating a gas at constant volume increases its pressure, which is essential in designing pressure vessels and engines.
Multivariable Calculus
Multivariable calculus extends the concepts of calculus to functions with more than one variable. It's essential for understanding how changes in one variable affect another in a multidimensional space. In the context of the ideal gas law, understanding partial derivatives, like \( \frac{\partial P}{\partial V} \) or \( \frac{\partial P}{\partial T} \), allows us to analyze how pressure changes independently when either volume or temperature changes. These derivatives offer valuable insights:
- \( \frac{\partial P}{\partial V} \): Shows the sensitivity of pressure to changes in volume.
- \( \frac{\partial P}{\partial T} \): Shows the sensitivity of pressure to changes in temperature.
Other exercises in this chapter
Problem 154
Find \(\lim _{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x}=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) for \(f(x, y)=x^{2} y
View solution Problem 155
Find \(\lim _{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x}=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) for \(f(x, y)=\sin (x
View solution Problem 157
The equation for heat flow in the \(x y\) -plane is \(\frac{\partial f}{\partial t}=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}
View solution Problem 158
The basic wave equation is \(f_{t t}=f_{x x}\). Verify that \(f(x, t)=\sin (x+t) \quad\) and \(\quad f(x, t)=\sin (x-t) \quad\) are solutions.
View solution