# How do I write Boyle's law as a power function?

Jul 27, 2015

You start by writing the mathematical expression for Boyle's Law.

#### Explanation:

Boyle's Law states that, for an ideal gas at constant temperature and number of moles, pressure and volume have an inverse relationship.

SImply put, when temperature and number of moles, i.e. the amount of gas, are held constant, increasing pressure will cause the volume to decrease, and vice versa.

Mathematically, you can write equation for Boyle's Law by starting from the ideal gas law equation.

$P V = n R T$, where

$P$ - the pressure of the gas;
$V$ - the volume it occupies;
$n$ - the number of moles of gas;
$R$ - the gas constant, equal to $0.082 \left(\text{atm" * "L")/("mol" * "K}\right)$
$T$ - th temperature of the gas.

If temperature and number of moles are kept constant, then the right side of the ideal gas law equation will be constant, since it's a product of three constant.

$P V = \text{constant} = k$

To write this as a power function, you need to get to the form

$f \left(x\right) = a \cdot {x}^{b}$, where

$a$, $b$ - constant real numbers.

Notice that if you isolate pressure on one side of the equation by dividing both sides by $V$, you can get

$\frac{P \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{V}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{V}}}} = \frac{k}{V}$

$P = \frac{k}{V}$

Since $\frac{1}{V}$ is equal to ${V}^{- 1}$, you get

$\textcolor{g r e e n}{P = k \cdot {V}^{- 1}} \to$ Boyle's Law as a power function

In your case, $a = k$ and $b = - 1$.

The inverse relationship between pressure and volume is confirmed by the $\left(- 1\right)$ exponent.