Problem 79 - Algebra and Trigonometry

Question

\(79-92\) Solve the equation for the indicated variable. $$ P V=n R T ; \quad \text { for } R $$

Step-by-Step Solution

Answer

Solve for \(R\) by rearranging the equation: \(R = \frac{PV}{nT}\).

Step 1: Identify the equation

We are given the equation \(PV = nRT\). The task is to solve this equation for the variable \(R\).

Step 2: Rewrite the equation

To solve for \(R\), we need to isolate \(R\) on one side of the equation. Begin with the given equation:\[PV = nRT\]

Step 3: Divide to isolate R

Divide both sides of the equation by \(nT\) to solve for \(R\):\[R = \frac{PV}{nT}\]

Step 4: Simplify the expression

Upon dividing, the equation simplifies directly to:\[R = \frac{PV}{nT}\]Now \(R\) is isolated and the equation is solved.

Key Concepts

Isolate the Variable

When you solve an equation, one of the most common techniques used is to isolate the variable you are solving for. This means you need to get the unknown variable on one side of the equation by itself. For example, in the equation provided, we are asked to solve for \( R \) in \( PV = nRT \). To isolate \( R \), our goal is to perform algebraic operations that remove the other variables from the side containing \( R \).

In this specific scenario, \( R \) is multiplied by \( nT \), so we perform the inverse operation, which is division, to remove \( nT \) from the right-hand side of the equation.

This is achieved by dividing both sides of the equation by \( nT \), leading to the variable \( R \) being isolated. Doing this keeps the equation balanced— an essential principle in algebra.

• Identify the variable you need to isolate.
• Use inverse operations to isolate the variable.
• Perform the same operation on both sides of the equation to maintain balance.

Solve Equation

Solving an equation is like piecing together a puzzle. You have to work step by step until you unveil the mystery of the unknown variable. In algebra, this often means tweaking the equation through various operations to reach the solution.

Take the equation \( PV = nRT \) we have in this task. Solving this equation begins with identifying what each component represents and knowing what operations can be applied. Here, since we need \( R \), we will manipulate the equation by using division to get \( R = \frac{PV}{nT} \). This is called solving the equation for \( R \).

This process involves a sequence of logical steps:

• Identify the equation and known variables.
• Apply the appropriate operations, such as addition, subtraction, multiplication, or division, to simplify the equation.
• Continue simplifying until the desired variable is isolated.
• Check your solution by substituting back into the original equation whenever possible.

Formula Rearrangement

In algebra, rearranging formulas is a crucial skill that allows us to solve equations for a specific variable, especially when dealing with formulas that describe real-world phenomena.

Consider the formula \( PV = nRT \). While it usually describes the relationship between pressure, volume, temperature, and other factors in chemistry (the ideal gas law), we can rearrange it to focus on any one variable. This is what we mean by 'formula rearrangement'.

To rearrange a formula:

• Identify the variable you need to solve for.
• Use algebraic operations to move other terms to the opposite side of the equation. Identify which operations will efficiently isolate your desired variable (e.g., division, multiplication).
• Simplify the expression if possible.

By rearranging \( PV = nRT \) to solve for \( R \), we systematically move each inherent component \( n \) and \( T \) to the opposite side through division, resulting in \( R = \frac{PV}{nT} \). This demonstrates formula rearrangement, allowing flexibility in solving for different variables depending on the scenario.