On Your ONn 7. Boyle's Law in chemistry states that when temperature is constant, the pressure of a gas is inversely proportional to its volume. Let \( p \) represent the pressure, and let \( V \) represent the volume of the gas. At a certain temperature, \( V=\frac{\text { constant }}{p} \). By what factor does the volume of the gas change if the pressure changes by each given factor? \( \begin{array}{ll}\text { a. } 2 & \text { b. } 3 \\ \text { c. } 4 & \text { d. } 5\end{array} \)

According to Boyle's Law, the relationship between pressure \( p \) and volume \( V \) of a gas at constant temperature can be expressed as: \[ V = \frac{k}{p} \] where \( k \) is a constant. This means that if the pressure changes, the volume will change inversely. Specifically, if the pressure changes by a factor of \( n \), the volume will change by a factor of \( \frac{1}{n} \). Let's analyze the changes in volume for each of the given factors of pressure: 1. **If the pressure changes by a factor of 2:** - New pressure = \( 2p \) - New volume = \( V' = \frac{k}{2p} = \frac{1}{2} \cdot \frac{k}{p} = \frac{1}{2} V \) - Volume changes by a factor of \( \frac{1}{2} \). 2. **If the pressure changes by a factor of 3:** - New pressure = \( 3p \) - New volume = \( V' = \frac{k}{3p} = \frac{1}{3} \cdot \frac{k}{p} = \frac{1}{3} V \) - Volume changes by a factor of \( \frac{1}{3} \). 3. **If the pressure changes by a factor of 4:** - New pressure = \( 4p \) - New volume = \( V' = \frac{k}{4p} = \frac{1}{4} \cdot \frac{k}{p} = \frac{1}{4} V \) - Volume changes by a factor of \( \frac{1}{4} \). 4. **If the pressure changes by a factor of 5:** - New pressure = \( 5p \) - New volume = \( V' = \frac{k}{5p} = \frac{1}{5} \cdot \frac{k}{p} = \frac{1}{5} V \) - Volume changes by a factor of \( \frac{1}{5} \). In summary, the volume of the gas changes by the following factors when the pressure changes: - For a pressure change by a factor of 2, the volume changes by a factor of \( \frac{1}{2} \). - For a pressure change by a factor of 3, the volume changes by a factor of \( \frac{1}{3} \). - For a pressure change by a factor of 4, the volume changes by a factor of \( \frac{1}{4} \). - For a pressure change by a factor of 5, the volume changes by a factor of \( \frac{1}{5} \). Thus, the volume of the gas decreases by the same factor as the increase in pressure.