Question 8.5: Home oxygen tanks can be dangerous if they are heated, becau...
Home oxygen tanks can be dangerous if they are heated, because they can explode. Suppose an oxygen tank has a pressure of 120 atm at a room temperature of 25 °C. If a fire in the room causes the temperature of the gas inside the oxygen tank to reach 402 °C, what is its pressure, in atmospheres, if the volume and amount of gas do not change? The oxygen tank may rupture if the pressure inside exceeds 180 atm. Would you expect it to rupture?
STEP 1 State the given and needed quantities .We place the gas data in a table by writing the initial temperature and pressure as T_{1} and P_{1} and the final temperature and pressure as T_{2} and P_{2} . We see that the temperature increases from 25 °C to 402 °C.
Using Gay-Lussac’s law,
\frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}} No change in volume and amount of gas (Gay-Lussac’s law)
we predict that the pressure increases.
T_{1} = 25 °C + 273 = 298 K
T_{2} = 402 °C + 273 = 675 K
| ANALYZE THE PROBLEM | Given | Need | Connect |
| P_{1} = 120 atm T_{1} = 298 K T_{2} = 675 K
Factors that do not change: V and n |
\boxed{P_{2}} | Gay-Lussac’s law, \frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}
Predict: T increases, P increases |
STEP 2 Rearrange the gas law equation to solve for the unknown quantity.
Using Gay-Lussac’s law, we solve for P_{2} by multiplying both sides by T_{2}.
\frac{P_{1}}{T_{1}} \times T_{2}=\frac{\boxed{P_{2} }}{\cancel{T_{2}}} \times \cancel{T_{2}}
\boxed{P_{2}} =P_{1} \times \frac{T_{2}}{T_{1}}
STEP 3 Substitute values into the gas law equation and calculate . When we substitute in the values, we see that the ratio of the temperatures (temperature factor) is greater than 1, which increases the pressure as predicted.
\boxed{P_{2}} = 120 atm \times \underset{\begin{array}{l}\text{Temperature factor}\\\text{increases pressure}\end{array}}{\frac{675 \cancel{K}}{298 \cancel{K}}} = 270 atmBecause the calculated pressure of 270 atm exceeds the limit of 180 atm, we would expect the oxygen tank to rupture.