Problem 17 - Algebra for College Students

Question

$$\text { Solve each formula for the indicated variable.}$$ $$\frac{P_{1} V_{1}}{T_{1}}=\frac{P_{2} V_{2}}{T_{2}} \text { for } T_{1}$$

Step-by-Step Solution

Answer

T_{1} = \frac{P_{1} V_{1} T_{2}}{P_{2} V_{2}}

Step 1: Write the given equation

Start with the given equation: \[ \frac{P_{1} V_{1}}{T_{1}} = \frac{P_{2} V_{2}}{T_{2}} \]

Step 2: Cross-multiply to eliminate fractions

Cross-multiply to move the denominators: \[ P_{1} V_{1} T_{2} = P_{2} V_{2} T_{1} \]

Step 3: Isolate the variable T_{1}

Solve for \( T_{1} \) by dividing both sides by \( P_{2} V_{2} \): \[ T_{1} = \frac{P_{1} V_{1} T_{2}}{P_{2} V_{2}} \]

Key Concepts

Ideal Gas Law

The ideal gas law is a cornerstone of thermodynamics and physical chemistry. It describes the relationship between pressure (P), volume (V), and temperature (T) of a gas. The basic form of the ideal gas law is:
\[ PV = nRT \]
where:

• P is the pressure of the gas.


• V is the volume of the gas.


• n is the number of moles of the gas.


• R is the gas constant.


• T is the temperature of the gas in Kelvin.

In the given exercise, a variation of the ideal gas law equation is used:
\[ \frac{P_{1} V_{1}}{T_{1}} = \frac{P_{2} V_{2}}{T_{2}} \]
This equation equates the state of the gas at two different conditions. Understanding the ideal gas law helps us predict how a gas will behave when conditions such as temperature, volume, or pressure change. Using this principle, we can solve for an unknown variable by manipulating the equation properly.

Cross-Multiplication

Cross-multiplication is a handy mathematical tool used to eliminate fractions in equations, making them easier to solve. Here is how cross-multiplication works:
We started with the equation:
\[ \frac{P_{1} V_{1}}{T_{1}} = \frac{P_{2} V_{2}}{T_{2}} \]
To get rid of the fractions, we cross-multiply, which means we multiply the numerator of one fraction by the denominator of the other fraction:
\[ P_{1} V_{1} \times T_{2} = P_{2} V_{2} \times T_{1} \]
This step simplifies the equation by getting rid of denominators, leaving us with:
\[ P_{1} V_{1} T_{2} = P_{2} V_{2} T_{1} \]
Now the equation is much easier to work with since we don’t have any fractions. Cross-multiplication is especially useful when solving proportions or equations involving ratios, as it straightens out the complexity and makes isolation of variables straightforward.

Isolating Variables

To solve for a specific variable in an equation, we need to isolate it on one side of the equation. This process involves algebraic manipulation. Let's walk through it with our example:
We have:
\[ P_{1} V_{1} T_{2} = P_{2} V_{2} T_{1} \]
We need to isolate \(T_{1}\). To do this, we divide both sides of the equation by \(P_{2} V_{2}\):
\[ \frac{P_{1} V_{1} T_{2}}{P_{2} V_{2}} = T_{1} \]
This gives us:
\[ T_{1} = \frac{P_{1} V_{1} T_{2}}{P_{2} V_{2}} \]
By isolating \(T_{1}\), we can see how changes in the other variables (\(P_{1}\), \(V_{1}\), \(T_{2}\), \(P_{2}\), and \(V_{2}\)) affect the temperature \(T_{1}\). Understanding how to isolate variables is crucial for solving many types of equations in physics, as it allows us to pinpoint and calculate specific unknowns.