Problem 29 - Precalculus Mathematics for Calculus

Question

Solve the equation for the indicated variable. $$P V=n R T ; \quad \text { for } R$$

Step-by-Step Solution

Answer

\( R = \frac{PV}{nT} \)

Step 1: Identify the equation

We start with the given equation, which is the Ideal Gas Law: \( PV = nRT \). We need to solve for \( R \).

Step 2: Isolate R

To solve for \( R \), we need to rearrange the equation. Divide both sides of the equation by \( nT \): \[ R = \frac{PV}{nT} \].

Step 3: Final expression for R

The expression \( R = \frac{PV}{nT} \) is the solution for \( R \), meaning \( R \) is isolated on one side of the equation.

Key Concepts

Equation Solving

Equation solving is the process of finding the unknown value in a mathematical equation. In this Ideal Gas Law problem, we have the equation \( PV = nRT \). Each letter in the equation represents a different physical property, making it crucial to solve for the variable of interest. Here, we are solving for \( R \), the ideal gas constant. The general approach to equation solving involves:

• Identifying the equation to work with.
• Determining which variable needs to be isolated.
• Rearranging the equation to express the unknown variable in terms of known quantities.

In solving equations, always ensure that your operations (addition, subtraction, multiplication, or division) maintain the balance of the equation. This means whatever you do to one side, you must also do to the other to keep the equation equal.Approaching equation solving step-by-step helps in understanding each operation's impact and guarantees accuracy in finding the correct value for the variable.

Isolate a Variable

Isolating a variable means rearranging an equation so that the variable of interest is alone on one side of the equation. This method is crucial when you need to find the value of a specific variable, as in the case of \( R \) in the Ideal Gas Law.When isolating a variable:

• Identify the variable you need to isolate. In our example, '\( R \)' is the target variable.
• Use inverse operations to move all other terms to the opposite side of the equation. This often involves basic operations such as addition, subtraction, multiplication, or division.
• Make sure each step is mathematically valid, maintaining equality throughout.

For \( PV = nRT \), isolating \( R \) involves dividing both sides by \( nT \) to get \( R = \frac{PV}{nT} \). This rearrangement succeeds in expressing \( R \) in terms of known quantities, thus isolating it completely.

Precalculus Mathematics

Precalculus mathematics forms the foundation for understanding advanced mathematical concepts by focusing on functions, calculations, and the essentials of algebra and trigonometry. These skills are vital when dealing with equations like the Ideal Gas Law. In the context of this exercise:

• Algebraic manipulation is used to rearrange equations and solve for desired variables.
• Understanding mathematical properties and operations ensures that rearrangements are done correctly without altering the equation's balance.
• Knowledge of functions and their inverses aids in solving equations involving multiple variables.

Understanding how to use precalculus mathematics efficiently helps in situations where multiple steps and operations are necessary, such as isolating variables and solving complex equations. The logical and methodical approach fostered by precalculus is crucial for mastering the solutions to equations like our Ideal Gas Law example.