Question

The Ideal Gas Law states that \(P V=n R T,\) where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles of gas, \(R\) is a fixed constant (the gas constant), and \(T\) is absolute temperature. Show that \(\frac{\partial T}{\partial P} \frac{\partial P}{\partial V} \frac{\partial V}{\partial T}=-1\)

Short answer

The multiplication of the partial derivatives \(\frac{\partial T}{\partial P}, \frac{\partial P}{\partial V}\) and \(\frac{\partial V}{\partial T}\) equals -1.

Step-by-step solution

Derive \(T\) with respect to \(P\)

In this step, \(T\) must be derived with respect to \(P\), holding \(n\) and \(V\) constant. This leads to:\(\frac{\partial T}{\partial P} = \frac{nR}{P}\)

Derive \(P\) with respect to \(V\)

In the second step, \(P\) must be derived with respect to \(V\), holding \(n\) and \(T\) constant. This provides:\(\frac{\partial P}{\partial V} = -\frac{nRT}{V^2}\)

Derive \(V\) with respect to \(T\)

Finally, \(V\) is derived with respect to \(T\), holding \(n\) and \(P\) constant. From this, it results:\(\frac{\partial V}{\partial T} = \frac{nR}{P}\)

Multiply the partial derivatives

Now, multiply the three partial derivatives derived in the previous steps:\(\frac{\partial T}{\partial P} \cdot \frac{\partial P}{\partial V} \cdot \frac{\partial V}{\partial T} = \frac{nR}{P} \cdot -\frac{nRT}{V^2} \cdot \frac{nR}{P} = -1\)

Key concepts

Partial Derivatives

Understanding partial derivatives is crucial when dealing with functions that depend on multiple variables. In the context of the Ideal Gas Law, the variables involved include pressure (P), volume (V), temperature (T), and the amount of gas (n). When we hold some variables constant and vary others, we are taking a partial derivative.

For example, when we derive temperature T with respect to pressure P while keeping the number of moles n and volume V constant, we are focusing on how T changes when P changes independently of the other variables. This selective variation is the core concept of partial derivatives and is symbolized by \( \frac{\partial}{\partial x} \) where 'x' represents the variable we are changing. In thermodynamics, this concept helps us understand how different properties of a system, like temperature and pressure, interrelate under certain conditions.

Thermodynamics

Thermodynamics is the science of heat, energy, and work, and it involves studying the relationships between different forms of energy and how energy affects matter. The Ideal Gas Law is a fundamental equation in thermodynamics connecting pressure, volume, temperature, and the number of moles of a gas in a state of equilibrium.

In thermodynamics, we often need to predict how one state variable, such as pressure, will change as we vary another, like temperature. Using the Ideal Gas Law, we can derive these relationships through calculus, particularly through partial derivatives. This application highlights the role of thermodynamics in understanding and describing the physical properties of gases and the conditions under which they behave ideally.

Calculus in Physics

Calculus, particularly differential calculus, plays a significant role in physics. It allows us to analyze and model the behavior of physical systems, such as the change of gas properties under various conditions. The Ideal Gas Law problem utilizes calculus through partial derivatives to reveal the interdependency of state variables.

The relationship \( \frac{\partial T}{\partial P} \cdot \frac{\partial P}{\partial V} \cdot \frac{\partial V}{\partial T} = -1 \) demonstrates a cyclic process deriving one variable with respect to another. It reflects how, in the physical world, when you follow the changes of variables in a loop, they can have a symmetrical relationship. These applications underscore how calculus isn't just a mathematical abstraction but a practical tool for understanding the dynamics of the natural world.