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]
A certain gas has a volume of 10 liters at a temperature of 404 Kelvins (K) and a pressure of 36 Pascals (Pa). If the temperature is raised to 450 K and the pressure is raised to 40 Pa, what is its volume?
To the nearest liter, the volume is $\square$ liters.
Answer (2)
The equation representing the relationship is P = k V T . After calculating with the new temperature and pressure, the volume is approximately 10 liters when rounded. Thus, the answer is 10 liters.
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The volume of a gas varies directly with temperature and inversely with pressure: V = k P T .
Find the constant k using the initial conditions: k = 101 90 .
Calculate the new volume V 2 using the new temperature and pressure: V 2 = 101 90 ⋅ 40 450 ≈ 10.02475 .
Round the new volume to the nearest liter: 10
Explanation
Understanding the Relationship The problem states that the volume of a gas, V , varies directly with temperature, T , and inversely with pressure, P . This means that V is proportional to T and inversely proportional to P . We can write this relationship as: V = k P T where k is a constant of proportionality.
Finding the Correct Equation The first part of the problem asks us to identify the equation that represents this relationship. We can rearrange the equation above to solve for P :
P = k V T So, the correct equation is P = k V T .
Calculating the Constant k Now, we are given that a certain gas has a volume of V 1 = 10 liters at a temperature of T 1 = 404 Kelvins (K) and a pressure of P 1 = 36 Pascals (Pa). We can use this information to find the constant k :
10 = k 36 404 Solving for k :
k = 10 ⋅ 404 36 = 404 360 = 101 90
Finding the New Volume Next, we are given that the temperature is raised to T 2 = 450 K and the pressure is raised to P 2 = 40 Pa. We want to find the new volume V 2 . Using the same relationship, we have: V 2 = k P 2 T 2 = 101 90 ⋅ 40 450 Calculating V 2 :
V 2 = 101 90 ⋅ 40 450 = 101 ⋅ 4 90 ⋅ 45 = 404 4050 ≈ 10.02475
Rounding to the Nearest Liter Finally, we need to round V 2 to the nearest liter. Since 10.02475 is very close to 10, we round it to 10 liters.
Examples
Understanding how gas volume changes with temperature and pressure is crucial in many real-world applications. For example, in designing engines, engineers need to predict how the volume of air-fuel mixture changes inside the cylinder as the temperature and pressure vary during the combustion cycle. Similarly, in meteorology, understanding the relationship between air volume, temperature, and pressure helps predict weather patterns. In medicine, ventilators use these principles to deliver the correct amount of air to patients based on their lung capacity and breathing rate. These principles are described by the ideal gas law, P V = n RT , which relates pressure, volume, and temperature of a gas.
