Problem 11 - Elementary Algebra

Question

Solve the equations and inequalities for the following problems. $$ \text { Solve } p=\frac{n R T}{V} \text { for } T $$

Step-by-Step Solution

Answer

Question: Solve the Ideal Gas Law equation for T, given the equation pV = nRT. Answer: T = (p * V) / (n * R)

Step 1: Isolate T on one side of the equation

To isolate T, we need to get rid of the other terms on the right side of the equation, namely n, R, and V.

Step 2: Multiply both sides of the equation by V

By multiplying both sides of the equation by V, we can cancel out the V from the denominator on the right side of the equation: $$ p * V = n * R * T $$

Step 3: Divide both sides of the equation by nR

Lastly, we need to get rid of n and R from the right side of the equation. To do this, we can divide both sides of the equation by nR: $$ \frac{p * V}{n * R} = T $$ Now, we have successfully isolated T and solved the equation: $$ T = \frac{p * V}{n * R} $$

Key Concepts

Isolating Variables

Mastering the art of isolating variables is crucial in algebra, as it allows you to solve for a specific unknown quantity within an equation. Let's explore the steps required for isolating variables, using the provided example.

When you're given an equation like \( p = \frac{n R T}{V} \) and asked to solve for \( T \) specifically, you are executing a process where \( T \) ends up being alone on one side of the equation. This process usually involves reversing whatever operations are affecting the variable you want to isolate. We often perform operations on both sides of the equation to maintain equality, which is the cornerstone principle of algebraic manipulations.

For instance, if a variable is being multiplied by some other terms, then you would typically divide both sides by those terms, and vice versa. It's similar to untangling a knot, where each move is deliberate to free one part without tightening another. In your example, multiplying by \( V \) and then dividing by \( nR \) were the key steps to unravel \( T \) from the other variables.

Manipulating Equations

Manipulating equations is a fundamental skill in algebra, as it involves performing a series of operations to reformulate the equation according to your needs, often to solve for a desired variable. The manipulation of equations must always follow the rules of algebra to ensure that the equality is preserved.

In the step-by-step solution where we aimed to isolate \( T \) on one side, two critical manipulations were made: multiplication and division. The multiplication by \( V \) canceled the denominator on the right side, while the division by \( nR \) separated \( T \) from the remaining product on the right side of the equation.

It's important to perform these actions with precision, ensuring that each operation is done to both sides of the equation. Skipping a step, or failing to apply an operation to both sides could lead to incorrect results. Remember, algebra is like a balancing act; whatever you do to one side, you do to the other, and that's how the balance is maintained throughout manipulations.

Algebraic Expressions

Algebraic expressions represent numerical relationships using variables, numbers, and operational symbols. Understanding these expressions is key to solving algebraic problems, as they form the language through which algebra communicates ideas.

In the exercise \( p = \frac{n R T}{V} \) for example, we see an algebraic expression that represents the mathematical relationship between pressure (\( p \) ), volume (\( V \) ) , the number of moles (\( n \) ), a constant (\( R \) ), and temperature (\( T \) ). This particular expression is derived from the ideal gas law in chemistry. To untangle this expression and solve for \( T \) solely, we strip away the other variables by performing inverse operations.

Getting familiar with these expressions empowers you to not only solve specific problems but also to formulate your own equations based on the relationships you observe — much like deciphering a code to reveal the message within.