Problem 83 - Algebra and Trigonometry

Question

Solve the equation for the indicated variable. \(P V=n R T ; \quad\) for \(R\)

Step-by-Step Solution

Answer

The solution for \(R\) is \( R = \frac{PV}{nT} \).

Step 1: Understand the equation

The given equation is the ideal gas law, represented as \(PV = nRT\). Here, \(P\) stands for pressure, \(V\) stands for volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. Our task is to solve this equation for the variable \(R\).

Step 2: Isolate the variable R

To isolate \(R\), we need to get \(R\) by itself on one side of the equation. Start by dividing both sides of the equation by \(nT\) to solve for \(R\). The solution to this rearrangement is: \[R = \frac{PV}{nT}.\]

Step 3: Verify the solution

Substitute back the derived formula \(R = \frac{PV}{nT}\) into the original equation to ensure it holds true when rearranged. When substituted back, it should simplify to the original equation, confirming the rearrangement is correct.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental concept in chemistry and physics, describing the behavior of an ideal gas. Its common formulation is \(PV = nRT\). The components of this equation are as follows:

• \(P\) represents the pressure of the gas.
• \(V\) is the volume the gas occupies.
• \(n\) stands for the number of moles of the gas.
• \(R\) is the ideal gas constant, a universal constant used in gas law calculations.
• \(T\) represents the temperature of the gas, typically measured in Kelvin.

This equation allows you to predict how changing one of these variables affects the others, assuming all the other conditions are ideal, meaning they don’t deviate from the typical behavior of gases. Understanding this formula is like having a key that can unlock the way gases react to changes in conditions like pressure or temperature. Remember, it simplifies calculations by assuming no loss of energy to the environment and that the molecules do not interact, which is not always true in real gases, but it makes modeling simpler.

Isolation of Variables

Isolation of variables involves rearranging an equation to solve for one specific variable, keeping it on one side while manipulating the other terms to be on the opposite side. In the ideal gas law example, our task was to solve for \(R\), the ideal gas constant. To do this, we need to move all other terms away from \(R\) using algebraic operations.
Start by identifying the target variable. For \(PV = nRT\) to solve for \(R\), you want \(R\) on one side.
By dividing each side of the equation by \(nT\), we isolate \(R\):\[R = \frac{PV}{nT}\]This means every operation performed on one side must be done to the other to maintain equality.
Isolation allows you to find exact relationships between variables or even predict one variable's impacts based on the rest. It is a foundational algebraic skill, crucial for working through complex equations across different areas of math and science.

Algebraic Manipulation

Algebraic manipulation includes a variety of techniques used to rearrange equations, factor expressions, and simplify terms. In our problem, isolating the variable \(R\) required using a basic form of algebraic manipulation: division.To manipulate the original formula \(PV = nRT\), we must understand the importance of performing inverse operations. In this case, multiplying is reversed by dividing. By dividing both sides by \(nT\), we precisely control the equation's structure:\[R = \frac{PV}{nT}\]Ensuring each operation is mirrored on each side of the equation keeps everything balanced, which is key in algebra.
Through manipulating equations, you gain insight, allowing transformations from complex to simpler forms, making equations more digestible for analysis or further calculations. This core skill is widely applicable in both academic settings and real-world problem-solving.