Question

The Effect of Altitude on the Lungs. (a) Calculate the change in air pressure you will experience if you climb a 1000-m mountain, assuming that the temperature and air density do not change over this distance and that they were 22°C and 1.2 kg/m³, respectively, at the bottom of the mountain. (Note: The result of Example 18.4 doesn’t apply, since the expression derived in that example accounts for the variation of air density with altitude and we are told to ignore that here.) (b) If you took a 0.50-L breath at the foot of the mountain and managed to hold it until you reached the top, what would be the volume of this breath when you exhaled it there?

Short answer

(a) The change in air pressure we will experience if we climb a 1000-m mountain will be $1.18x{10}^4\mathrm{Pa}$.

(b) If you took a 0.50-L breath at the foot of the mountain and managed to hold it until you reached the top, the volume of this breath when you exhaled it there will be 0.566 L.

Step-by-step solution

Step 1:

When you climb a mountain there will be a change in pressure decreased by∆p, this change in pressure affected by the density of air$\rho$, the gravity acceleration g and definitely the height h of the mountain. and this relation employed by

$\mathbf{∆}\mathbf{p}\mathbf{=}\mathbf{\rho gh}$

where$\rho$is given by 1.2 kg/ height h is given by 1000 m.and the negative sign represent the decrease in pressure.

$\begin{array}{r}\mathrm{\Delta }p=-pgh\\ =-1.2\mathrm{kg}/{\mathrm{m}}^3\times 9.8\mathrm{m}/{\mathrm{s}}^2\times 1000\mathrm{m}\\ =-1.18\times {10}^4\mathrm{Pa}\end{array}$

After climb mountain with height 1000 m, the pressure applied to your body changed byrole="math" localid="1668231600961" $1.18\times {10}^4\mathrm{Pa}$

Note: We neglect the effect of the change in density and temperature as given in the problem

Step 2:

The pressure at the bottom is the atmospheric pressure which equals$10.13x{10}^4\mathrm{Pa}$

In Step 1 we calculated the change in pressure after climbing the mountain so we can calculate the pressure p, after climbing the mountain by added the value of the change in pressure∆p to p₁ and would employ by

$\begin{array}{r}{p}_2=p_1+\mathrm{\Delta }p\\ =10.13\times {10}^4\mathrm{Pa}+\left(-1.18\times {10}^4\mathrm{Pa}\right)\\ =8.95\times {10}^4\mathrm{Pa}\end{array}$

Thus, the change in air pressure you will experience if you climb a 1000-m mountain is$8.95\times {10}^4\mathrm{Pa}$

Step 3:

To find the volume V₂ of the breath at the top of the mountain, we want to find a relation between V and p and we could get it from Ideal gas law

pV = nRT

where$p\infty \frac{1}{V}$ and the relation between would be given by

$\begin{array}{r}{p}_1{V}_1=p_2{V}_2\\ V_2=\frac{p_1{V}_1}{p_2}\\ =\frac{10.13\times {10}^4\mathrm{Pa}\times 0.50\mathrm{L}}{8.95\times {10}^4\mathrm{Pa}}\\ =0.566\mathrm{L}\end{array}$

∴ If you took a 0.50-L breath at the foot of the mountain and managed to hold it until you reached the top, the volume of this breath when you exhaled it there will be 0.566 L.