The volume of a gas varies directly with temperature, [tex]$T$[/tex], and inversely with pressure, [tex]$P$[/tex]. Which of the following equations represents that relationship?
[tex]$P=k T V$[/tex]
[tex]$P=k \frac{T}{V}$[/tex]
[tex]$P=k \frac{V}{T}$[/tex]
[tex]$P=k \frac{1}{T V}$[/tex]
Answer (1)
Volume varies directly with temperature: V ∝ T .
Volume varies inversely with pressure: V ∝ P 1 .
Combine proportionalities: V ∝ P T , then V = k P T .
Express P in terms of V and T : P = k V T .
Explanation
Understanding the Problem We are given that the volume of a gas, V , varies directly with temperature, T , and inversely with pressure, P . This means that as the temperature increases, the volume increases proportionally, and as the pressure increases, the volume decreases proportionally. We need to find the equation that represents this relationship from the given options.
Combining Proportionalities Since V varies directly with T , we can write V ∝ T . Since V varies inversely with P , we can write V ∝ P 1 . Combining these two proportionalities, we get V ∝ P T .
Introducing the Constant of Proportionality To convert the proportionality into an equation, we introduce a constant of proportionality, k . Thus, we have V = k P T .
Rearranging the Equation Now, we need to rearrange the equation to express P in terms of V and T . Multiplying both sides by P , we get P V = k T . Dividing both sides by V , we get P = k V T .
Finding the Correct Option Comparing our result with the given options, we see that the correct equation is P = k V T .
Examples
Understanding the relationship between pressure, volume, and temperature of a gas is crucial in many real-world applications. For example, in internal combustion engines, the pressure and temperature of the gas inside the cylinder change as the volume changes during the piston's movement. This relationship is also important in weather forecasting, where changes in air pressure and temperature can affect the volume of air masses, leading to various weather phenomena. By understanding the equation P = k V T , engineers and scientists can design more efficient engines and make more accurate weather predictions.
