Think about the analogy "full of hot air." If a person were to inhale 2.2 L of gas at a temperature of 18 °C and that air heats to a temperature of 38 °C inside the lungs, what would be the new volume of the gas? Round to the nearest hundredth. Don't forget the units.
To solve this problem, we can use the Ideal Gas Law equation, which states: PV = nRT.
P = pressure (which we can assume is constant),
V = volume (which is what we are trying to find),
n = number of moles of gas,
R = ideal gas constant, and
T = temperature in Kelvin.
First, let's convert the temperatures from Celsius to Kelvin. To do this, we add 273.15 to each temperature value.
Temperature in Kelvin:
Initial temperature (T1) = 18 + 273.15 = 291.15 K
Final temperature (T2) = 38 + 273.15 = 311.15 K
Next, let's use the given information to find the initial number of moles of gas inhaled. We can use the ideal gas equation rearranged to solve for n:
n = PV / RT
Assuming the pressure (P) and the ideal gas constant (R) remain constant, we can calculate n using the initial condition and given volume:
2.2 L * P / (R * T1)
Since we don't have the exact value for the pressure, we can assume it cancels out when we calculate the ratio of the volumes. Therefore, we don't need to consider the pressure factor. We can proceed directly to calculating the new volume.
Using the equation for the final volume (V2), we can find the new volume of gas:
V2 = (n * R * T2) / P
We already calculated the value for n*R from the initial condition, so we substitute that into the equation:
V2 = (2.2 * R * T2) / P
Now, we need to find the value of R. The ideal gas constant, R, can be found using the relationship between the pressure, volume, and temperature of one mole of an ideal gas at STP (Standard Temperature and Pressure). At STP, 1 mole of an ideal gas occupies 22.4 L at a pressure of 1 atmosphere (atm) and a temperature of 273.15 K.
So, we can calculate R using the equation:
R = (P * V) / (n * T)
Plugging in the values for a mole of gas at STP:
R = (1 atm * 22.4 L) / (1 mole * 273.15 K)
Now we have all the information needed to calculate the new volume (V2) of the gas:
V2 = (2.2 * (P * V) / (T1 * 273.15 K) * (T2 - 273.15 K)) / P
Here, the pressure (P) cancels out, so we can simplify the equation to:
V2 = V * (T2 - 273.15 K) / T1
Now you can plug in the values:
V2 = 2.2 L * (311.15 K - 273.15 K) / 291.15 K
Simplifying the equation:
V2 = 2.2 L * 0.1303
V2 = 0.2867 L
Therefore, the new volume of gas would be approximately 0.29 L (rounded to the nearest hundredth).