Algebra Examples

$\ln (\frac{P_{2}}{P_{1}}) = - \frac{H}{R} \cdot (\frac{1}{T_{2}} - \frac{1}{T_{1}})$

Step 1

To solve for $P_{1}$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{P_{2}}{P_{1}})} = e^{- \frac{H}{R} \cdot (\frac{1}{T_{2}} - \frac{1}{T_{1}})}$

Step 2

Rewrite $\ln (\frac{P_{2}}{P_{1}}) = - \frac{H}{R} \cdot (\frac{1}{T_{2}} - \frac{1}{T_{1}})$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- \frac{H}{R} \cdot (\frac{1}{T_{2}} - \frac{1}{T_{1}})} = \frac{P_{2}}{P_{1}}$

Step 3

Solve for $P_{1}$ .

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Step 3.1

Rewrite the equation as $\frac{P_{2}}{P_{1}} = e^{- \frac{H}{R} \cdot (\frac{1}{T_{2}} - \frac{1}{T_{1}})}$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H}{R} \cdot (\frac{1}{T_{2}} - \frac{1}{T_{1}})}$

Step 3.2

Simplify $e^{- \frac{H}{R} \cdot (\frac{1}{T_{2}} - \frac{1}{T_{1}})}$ .

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Step 3.2.1

Apply the distributive property.

$\frac{P_{2}}{P_{1}} = e^{- \frac{H}{R} \cdot \frac{1}{T_{2}} - \frac{H}{R} (- \frac{1}{T_{1}})}$

Step 3.2.2

Multiply $\frac{1}{T_{2}}$ by $\frac{H}{R}$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H}{T_{2} R} - \frac{H}{R} (- \frac{1}{T_{1}})}$

Step 3.2.3

Multiply $- \frac{H}{R} (- \frac{1}{T_{1}})$ .

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Step 3.2.3.1

Multiply $- 1$ by $- 1$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H}{T_{2} R} + 1 \frac{H}{R} \frac{1}{T_{1}}}$

Step 3.2.3.2

Multiply $\frac{H}{R}$ by $1$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H}{T_{2} R} + \frac{H}{R} \cdot \frac{1}{T_{1}}}$

Step 3.2.3.3

Multiply $\frac{H}{R}$ by $\frac{1}{T_{1}}$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H}{T_{2} R} + \frac{H}{R T_{1}}}$

Step 3.3

Factor each term.

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Step 3.3.1

To write $- \frac{H}{T_{2} R}$ as a fraction with a common denominator, multiply by $\frac{T_{1}}{T_{1}}$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H}{T_{2} R} \cdot \frac{T_{1}}{T_{1}} + \frac{H}{R T_{1}}}$

Step 3.3.2

To write $\frac{H}{R T_{1}}$ as a fraction with a common denominator, multiply by $\frac{T_{2}}{T_{2}}$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H}{T_{2} R} \cdot \frac{T_{1}}{T_{1}} + \frac{H}{R T_{1}} \cdot \frac{T_{2}}{T_{2}}}$

Step 3.3.3

Write each expression with a common denominator of $T_{2} R T_{1}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.3.1

Multiply $\frac{H}{T_{2} R}$ by $\frac{T_{1}}{T_{1}}$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H T_{1}}{T_{2} R T_{1}} + \frac{H}{R T_{1}} \cdot \frac{T_{2}}{T_{2}}}$

Step 3.3.3.2

Multiply $\frac{H}{R T_{1}}$ by $\frac{T_{2}}{T_{2}}$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H T_{1}}{T_{2} R T_{1}} + \frac{H T_{2}}{R T_{1} T_{2}}}$

Step 3.3.3.3

Reorder the factors of $T_{2} R T_{1}$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H T_{1}}{T_{2} T_{1} R} + \frac{H T_{2}}{R T_{1} T_{2}}}$

Step 3.3.3.4

Reorder the factors of $R T_{1} T_{2}$ .

$\frac{P_{2}}{P_{1}} = e^{- \frac{H T_{1}}{T_{2} T_{1} R} + \frac{H T_{2}}{T_{2} T_{1} R}}$

Step 3.3.4

Combine the numerators over the common denominator.

$\frac{P_{2}}{P_{1}} = e^{\frac{- H T_{1} + H T_{2}}{T_{2} T_{1} R}}$

Step 3.3.5

Factor $H$ out of $- H T_{1} + H T_{2}$ .

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Step 3.3.5.1

Factor $H$ out of $- H T_{1}$ .

$\frac{P_{2}}{P_{1}} = e^{\frac{H (- T_{1}) + H T_{2}}{T_{2} T_{1} R}}$

Step 3.3.5.2

Factor $H$ out of $H T_{2}$ .

$\frac{P_{2}}{P_{1}} = e^{\frac{H (- T_{1}) + H (T_{2})}{T_{2} (T_{1}) R}}$

Step 3.3.5.3

Factor $H$ out of $H (- T_{1}) + H (T_{2})$ .

$\frac{P_{2}}{P_{1}} = e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}$

Step 3.4

Find the LCD of the terms in the equation.

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Step 3.4.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$P_{1}, 1$

Step 3.4.2

The LCM of one and any expression is the expression.

$P_{1}$

Step 3.5

Multiply each term in $\frac{P_{2}}{P_{1}} = e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}$ by $P_{1}$ to eliminate the fractions.

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Step 3.5.1

Multiply each term in $\frac{P_{2}}{P_{1}} = e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}$ by $P_{1}$ .

$\frac{P_{2}}{P_{1}} P_{1} = e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}} P_{1}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $P_{1}$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{P_{2}}{P_{1}} P_{1} = e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}} P_{1}$

Step 3.5.2.1.2

Rewrite the expression.

$P_{2} = e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}} P_{1}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Reorder factors in $e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}} P_{1}$ .

$P_{2} = P_{1} e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}$

Step 3.6

Solve the equation.

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Step 3.6.1

Rewrite the equation as $P_{1} e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}} = P_{2}$ .

$P_{1} e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}} = P_{2}$

Step 3.6.2

Divide each term in $P_{1} e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}} = P_{2}$ by $e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}$ and simplify.

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Step 3.6.2.1

Divide each term in $P_{1} e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}} = P_{2}$ by $e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}$ .

$\frac{P_{1} e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}}{e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}} = \frac{P_{2}}{e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}}$

Step 3.6.2.2

Simplify the left side.

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Step 3.6.2.2.1

Cancel the common factor of $e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}$ .

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Step 3.6.2.2.1.1

Cancel the common factor.

$\frac{P_{1} e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}}{e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}} = \frac{P_{2}}{e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}}$

Step 3.6.2.2.1.2

Divide $P_{1}$ by $1$ .

$P_{1} = \frac{P_{2}}{e^{\frac{H (- T_{1} + T_{2})}{T_{2} T_{1} R}}}$