Problem 46 - Intermediate Algebra

Question

Solve each formula for the specified variable. \(\frac{P_{1} V_{1}}{T_{1}}=\frac{P_{2} V_{2}}{T_{2}}\) for \(T_{2}\) (from chemistry)

Step-by-Step Solution

Answer

\( T_2 = \frac{P_2 V_2 T_1}{P_1 V_1} \)

Step 1: Cross-Multiply to Eliminate Fractions

Start by cross-multiplying to eliminate the fractions. Multiply both sides of the equation by \(T_2\) and \(T_1\) to have: \[ P_1 V_1 T_2 = P_2 V_2 T_1 \].

Step 2: Isolate T_2

To solve for \(T_2\), divide both sides of the equation by \(P_1 V_1\). This will give you \(T_2\) isolated on one side: \[ T_2 = \frac{P_2 V_2 T_1}{P_1 V_1} \].

Key Concepts

Cross-Multiplication

Cross-multiplication is a fundamental technique used in mathematics to solve equations that involve fractions. It simplifies the process by eliminating the fractions, allowing you to work with whole numbers or variables instead. In essence, when you have an equation set up like a proportion, \[\frac{a}{b} = \frac{c}{d}\] you can cross-multiply to eliminate the denominators. This involves multiplying the numerator of one side by the denominator of the other side.

• Multiply \(a\) by \(d\) to get \(a \cdot d\).
• Multiply \(b\) by \(c\) to get \(b \cdot c\).

By setting these two products equal,\( a \cdot d = b \cdot c \), you have effectively removed the fractions and simplified the equation to a form that is often easier to solve.This technique was used in our original problem to convert the equation from a fraction form \(\frac{P_{1} V_{1}}{T_{1}} = \frac{P_{2} V_{2}}{T_{2}}\), into the format \(P_1 V_1 T_2 = P_2 V_2 T_1\). Cross-multiplication is particularly helpful in problems involving direct and inverse variation or in converting units in chemistry-related equations.

Solving for a Variable

Solving for a variable means manipulating an equation in such a way that you isolate the variable you're interested in. This involves using algebraic operations to rewrite the equation so your variable appears on one side with everything else on the other.For instance, after cross-multiplying in our original equation, we obtained:\[ P_1 V_1 T_2 = P_2 V_2 T_1 \]The goal is to isolate \(T_2\) on one side of the equation. The most common method is to use division to move all the other factors to the opposite side.Here’s what you do:

• Divide both sides of the equation by \(P_1 V_1\).
• This effectively "undoes" the multiplication by \(P_1 V_1\) on the left side.

This gives:\[ T_2 = \frac{P_2 V_2 T_1}{P_1 V_1} \]Isolating a variable makes it easier to see relationships and solve for particular situations, crucial in various applications of physics and chemistry.

Chemical Formulas

Chemical formulas are symbolic representations of a compound that display the types and numbers of atoms present. They also hold significant value in chemistry when calculating things like pressure, volume, and temperature, as seen with the ideal gas law in the problem.In calculations, chemical formulas provide essential quantities needed to predict the behavior of substances under different conditions. Our problem involved such concepts but was abstracted to general terms \(P\), \(V\), and \(T\) which stand for pressure, volume, and temperature, respectively. These formulas are crucial when performing calculations related to chemical reactions and states of matter. Understanding chemical formulas helps us:

• Predict the properties of substances.
• Understand how different compounds react.
• Perform conversions between different states of matter and conditions such as temperature and pressure changes.

By developing a solid grasp of chemical formulas, students can better approach real-world chemistry problems by applying mathematical concepts like cross-multiplication and variable isolation.