Question

If \(p=\frac{R T}{V}\) where \(R\) is a constant, find \(\frac{\partial p}{\partial V}\) and \(\frac{\partial p}{\partial T}\)

Short answer

Question: Calculate the partial derivatives of the pressure equation \(p = \frac{RT}{V}\) with respect to volume (V) and temperature (T). Answer: The partial derivatives are \(\frac{\partial p}{\partial V} = -RTV^{-2}\) and \(\frac{\partial p}{\partial T} = RV^{-1}\).

Step-by-step solution

Apply the power rule to \(\frac{1}{V}\)

To make the differentiation easier, rewrite the equation as \(p = RTV^{-1}\), where \(V\) has an exponent of -1.

Differentiate with respect to \(V\)

Now, differentiate the equation with respect to \(V\), keeping in mind that \(R\) and \(T\) are constants. The result is: \(\frac{\partial p}{\partial V} = -RT(-1)V^{-1-1} = -RTV^{-2}\) Now, we will find \(\frac{\partial p}{\partial T}\).

Recall \(p=\frac{RT}{V}\)

The equation given was \(p = \frac{R T}{V}\). Since we are finding the derivative with respect to \(T\), we don't need to rewrite the equation.

Differentiate with respect to \(T\)

Differentiate the equation with respect to \(T\), keeping in mind that \(R\) and \(V\) are constants. The result is: \(\frac{\partial p}{\partial T} = R\frac{\partial T}{\partial T}V^{-1} = RV^{-1}\) In conclusion, the two partial derivatives are: \(\frac{\partial p}{\partial V} = -RTV^{-2}\) \(\frac{\partial p}{\partial T} = RV^{-1}\)

Key concepts

Mathematics for Engineers

Mathematics is a foundational element in engineering, providing tools and frameworks for analyzing and solving problems across various disciplines of engineering. One such tool is calculus, which is utilized to understand and calculate changes in physical quantities. Engineers often deal with systems where multiple variables interact, necessitating the use of partial differentiation to investigate how changes in one variable affect the system while others remain constant.

For instance, let’s consider the formula for pressure in a gas, where mathematically-defined relationships between volume, temperature, and pressure can help engineers predict how a gas might behave under different conditions. By mastering these concepts, engineers can design more efficient machinery, predict material behavior under stress, or optimize processes such as heating and cooling systems.

Differentiation with Respect to a Variable

Differentiation refers to the mathematical process of finding the rate at which a function changes at any point. When dealing with functions of multiple variables, partial differentiation comes into play, allowing us to focus on the change in one variable while holding others constant. This is a core principle when analyzing systems with interdependent variables, which is common in engineering problems.

In the given exercise, we differentiated the pressure, represented by the function p, with respect to volume (V) and temperature (T), examining how pressure changes as each of these variables changes independently. Through partial differentiation, engineers can isolate and study the effects of singular components within a system, aiding in more precise predictions and adjustments in their field.

Power Rule for Differentiation

The power rule is a basic yet powerful tool for differentiation, especially when dealing with polynomial functions or variables raised to a power. The rule states that if you have a function of the form f(x) = x^n, the derivative f'(x) is nx^(n-1). This simplifies the process of finding rates of change and is especially useful when the function involves a variable raised to a power, such as in the case of the inverse relationship between pressure and volume.

In our solution, we applied the power rule for differentiation to the term V^-1, treating it as a variable raised to the power of -1. By doing so, we could easily calculate how a slight change in volume results in a change in pressure. Understanding the power rule enables students to tackle a wide array of problems in mathematics, physics, and engineering, where variables often interact through exponential relationships.