Problem 77 - Intermediate Algebra

Question

Solve for the specified variable. $$ P V=n r t \quad \text { for } r $$

Step-by-Step Solution

Answer

\( r = \frac{PV}{nt} \)

Step 1: Understand the equation format

The equation given is a version of the ideal gas law: \( PV = nrt \). Our task is to solve this equation for the variable \( r \). Here, \( P \) is pressure, \( V \) is volume, \( n \) is the amount of substance (in moles), \( r \) is the ideal gas constant, and \( t \) is temperature.

Step 2: Isolate the variable

To isolate \( r \) on one side of the equation, we need to divide both sides by \( nt \), so that \( r \) will be alone. Perform the division as follows: \[\frac{PV}{nt} = r\] Thus, \( r \) is separated on one side of the equation.

Step 3: Result Interpretation

Now, we have successfully expressed \( r \) in terms of the other variables in the equation. This step confirms that \( r = \frac{PV}{nt} \), which is the expression of the ideal gas constant in terms of pressure, volume, number of moles, and temperature.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics, describing the relationship between pressure, volume, temperature, and the amount of gas. This law is crucial for understanding how gases behave under different conditions.

The Ideal Gas Law is represented by the equation:

• \( PV = nRT \)

Where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume occupied by the gas.
• \( n \) denotes the number of moles of the gas.
• \( R \) is the ideal gas constant, a proportionality factor.
• \( T \) is the absolute temperature, measured in Kelvin.

The equation indicates that for a given quantity of gas, increasing the pressure will decrease the volume, provided the temperature remains unchanged. Similarly, increasing the temperature will increase the volume if pressure is constant. Understanding these relationships helps in predicting how gases will behave in different scenarios.

Algebraic Manipulation

In mathematics, algebraic manipulation is a process used to rearrange equations and make them easier to solve. It involves applying mathematical operations to simplify expressions or isolate specific variables.

When solving equations, manipulating the equation by adding, subtracting, multiplying, or dividing both sides by the same number is crucial. For instance, in the Ideal Gas Law equation \( PV = nRT \), we wanted to isolate the variable \( R \). To do this, the equation was manipulated by dividing both sides by \( nT \).

• Divide both sides by \( nT \): \( \frac{PV}{nT} = R \)

This kind of manipulation retains equality on both sides of the equation, a critical step in solving for unknowns. Proper understanding of algebraic manipulation ensures that equation solving is straightforward and accurate.

Variable Isolation

Variable isolation is a key technique used in algebra to solve equations involving unknowns. This technique aims to express the variable of interest alone on one side of an equation, allowing its value to be determined.

To isolate a variable, you need to undo operations performed on this variable. In our exercise, we isolated the variable \( R \) in the equation \( PV = nRT \). This involved dividing by the other terms associated with \( R \), specifically \( nT \).

Steps for isolating a variable:

• Identify the variable to isolate (e.g., \( R \)).
• Perform operations to move other terms to the opposite side (e.g., divide by \( nT \)).
• Ensure the variable stands alone on one side of the equation, resulting in \( R = \frac{PV}{nT} \).

By isolating variables, solving complicated equations becomes more manageable, and it allows us to express one quantity in terms of others, as seen in the Ideal Gas Law example.