On a chilly 10^∘ C day, you quickly take a deep breath-all...

On a chilly $10^{\circ} \mathrm{C}$ day, you quickly take a deep breath-all your lungs can hold, $4.0 \mathrm{L}$. The air warms to your body temperature of $37^{\circ} \mathrm{C}$. If the air starts at a pressure of $1.0 \mathrm{atm},$ and you hold the volume of your lungs constant (a good approximation) and the number of molecules in your lungs stays constant as well (also a good approximation), what is the increase in pressure inside your lungs?

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Key Concepts

Absolute Temperature (Kelvin Conversion)

Temperature in the ideal gas law must be expressed in Kelvin because it is an absolute temperature scale. Converting temperatures from Celsius to Kelvin (by adding 273.15) ensures that the relations derived from the ideal gas law are correctly scaled and that the proportionalities involved remain valid.

Ideal Gas Law

The ideal gas law (PV = nRT) is the fundamental equation that relates the pressure, volume, amount of substance, and temperature of a gas. In problems involving changes in temperature, volume, or pressure, this law allows us to predict how one variable will change when others are held constant. Here, since the volume and the number of moles are constant, the change in pressure can be directly related to the change in absolute temperature.

Pressure-Temperature Relationship at Constant Volume

For a fixed volume and constant moles of gas, the pressure is directly proportional to the absolute temperature (P ? T). This relationship, sometimes referred to in this context as Gay-Lussac's law, means that any increase in the temperature of the gas results in a proportional increase in pressure, as long as the volume does not change.

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