Problem 57 - Elementary Algebra

Question

For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables. Solve \(p=\frac{n R T}{V}\) for \(R\).

Step-by-Step Solution

Answer

Question: Given the equation \(p = \frac{n R T}{V}\), solve for \(R\). Answer: \(R = \frac{pV}{nT}\)

Step 1: Write down the given equation

We have the equation: \(p = \frac{n R T}{V}\).

Step 2: Isolate the term containing \(R\)

To isolate the term containing \(R\), multiply both sides of the equation by \(V\): \(pV = n R T\).

Step 3: Solve for \(R\)

Now, divide both sides of the equation by \(nT\) to solve for \(R\): \(R = \frac{pV}{nT}\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a foundational skill in mathematics, particularly in solving equations. It involves performing operations on both sides of an equation to rearrange or simplify it. These operations must maintain the balance of the equation — meaning whatever you do to one side, you must do to the other.

In the context of the given exercise, where we have the gas law equation, algebraic manipulation includes operations like multiplication and division to isolate the desired variable. This process enables us to rewrite equations in a more useful form, depending on what quantity we are interested in solving for.

Some key techniques of algebraic manipulation include:

• Adding or subtracting the same amount from both sides of an equation to maintain equality.
• Multiplying or dividing both sides of an equation by the same non-zero number.
• Expanding brackets and simplifying expressions to make isolation of a variable more straightforward.
• Factoring to reduce equations into simpler forms.

Understanding and applying these techniques allows us to manipulate and solve even complex literal equations efficiently.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable of interest is by itself on one side of the equation, with all other terms on the opposite side. This process is crucial in solving literal equations, where the goal is to express a specific variable in terms of the others.

In our exercise, the variable 'R' (representing the ideal gas constant) needs to be isolated in the gas law equation. This requires that all other variables and constants be moved to the other side of the equal sign.

To isolate 'R', one must:

• Recognize the terms that are connected to 'R'.
• Perform inverse operations to detach 'R' from those terms.
• Remember to perform each operation on both sides of the equation.

For example, if 'R' is multiplied by another variable, we can isolate it by dividing both sides by that variable, thus 'undoing' the multiplication. This step-by-step approach ensures that the variable being solved for is clearly and cleanly presented, making it easier to interpret or use in further calculations.

Gas Law Equation

The gas law equation represented by the initial equation in our exercise, \(p = \frac{n R T}{V}\), is a form of the Ideal Gas Law. The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure (\(p\)), volume (\(V\)), temperature (\(T\)), and amount of substance (\(n\)) of an ideal gas to the gas constant (\(R\)).

The equation for solving 'R' derived from the Ideal Gas Law is useful in various scientific calculations where it's necessary to determine the gas constant based on known quantities of pressure, volume, temperature, and moles of the gas.

The Ideal Gas Law can be rearranged to solve for any one of its variables, depending on what information is available and what needs to be found. The ability to manipulate this equation is vital for scientists and engineers who frequently work with gases. As simplicity is key in solving such formulae, being sure to understand each component:

• \(p\) stands for 'pressure' of the gas.
• \(V\) is the 'volume' that the gas occupies.
• \(n\) represents the 'amount of substance' in moles.
• \(T\) is the 'temperature' in Kelvin.
• \(R\) is the 'ideal gas constant'.

Selecting and isolating the desired variable from the Ideal Gas Law requires algebraic manipulation, which students can master with practice, enabling them to tackle a wide variety of thermodynamic problems.