Question

A gas mixture with a total pressure of 745 \(\mathrm{mm} \mathrm{Hg}\) contains each of these gases at the indicated partial pressures: \(\mathrm{CO}_{2}, 125 \mathrm{mmHg} ; \mathrm{Ar}, 214 \mathrm{mmHg}\) ; and \(\mathrm{O}_{2}, 187 \mathrm{mmHg}\) . The mixture also contains helium gas. What is the partial pressure of the helium gas? What mass of helium gas is present in a 12.0 -L sample of this mixture at 273 \(\mathrm{K}\) ?

Short answer

The partial pressure of the helium gas is 219 mmHg, and the mass of helium gas present in a 12.0 L sample of this mixture at 273 K is 1.96 grams.

Step-by-step solution

Calculate Partial Pressure of Helium

Begin by calculating the partial pressure of helium (He) using Dalton's Law of Partial Pressures. Dalton's Law states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each individual gas in the mixture. The partial pressure of helium can be found by subtractng the sum of the partial pressures of the other gases from the total pressure. The formula is: Partial Pressure of He = Total Pressure - Sum of other partial pressures.

Use the Ideal Gas Law to Calculate the Moles of Helium

Once you have the partial pressure of helium, use the Ideal Gas Law, which is given by the equation PV = nRT, where P is pressure, V is volume, n is the number of moles of the gas, R is the gas constant (0.0821 \(\mathrm{L}\cdot\mathrm{mol}^{-1}\cdot\mathrm{K}^{-1}\)), and T is temperature in Kelvin, to find the number of moles of helium. Rearrange the equation to solve for n (moles of helium): n = PV/(RT). Make sure to convert the pressure of helium from mmHg to atm when using the Ideal Gas Law.

Calculate the Mass of Helium

The final step is to convert the number of moles of helium to mass using the molar mass of helium. The molar mass of helium (He) is about 4.00 \(\mathrm{g/mol}\). Use the formula: Mass of He = n (moles of He) * Molar Mass of He.

Key concepts

Dalton's Law of Partial Pressures

When you open a can of soda and hear the fizz, that's actually Dalton's Law of Partial Pressures in action! Imagine each hiss is a different gas escaping — that's like the gases in our problem trying to tell us how much they contribute to the overall 'fizz' of the total pressure. Dalton's Law tells us that in a blend of gases, each one's partial pressure adds up to create the total pressure you feel in that can of soda, or in this case, a gas mixture.

Simply put, if you know the individual pressures of all but one of the gases in a mixture, you can figure out the missing one by subtracting the sum of the known pressures from the total pressure. Kind of like solving a mystery by ruling out all the suspects you know didn't do it!

Ideal Gas Law

If gases had a superhero league, the Ideal Gas Law would be their rulebook. It's a fundamental equation that describes how gases behave if we assume they're 'ideal,' meaning they don't attract or repel each other and take up no space. While no gas is truly ideal, many gases act close enough to it that we can use the law to make some pretty good predictions.

The equation for the Ideal Gas Law is \( PV = nRT \), making it easy to juggle the four key players in the gas game: pressure (P), volume (V), moles of gas (n), and temperature (T). The R is just the referee making sure the game is played fairly — it's a constant value that keeps the units in check. So, when we know any three of these variables, we can use them to find the fourth. It's like knowing three ingredients in a secret sauce and using the recipe (the law) to figure out the fourth to make your dish just right.

Molar Mass of Helium

Think of helium as the light-weight champion in the world of elements. Its molar mass, which is just a fancy way of saying how much one mole (think 'bag') of helium weighs, is around 4.00 grams. That's incredibly light, which is why balloons filled with helium soar high into the sky. The molar mass acts like a conversion rate, showing us how to change moles, which measure the amount of particles, into grams, which most of us understand as weight.

When we're looking at the mass of helium in a gas mixture, we're really asking, 'How many balloons' worth of helium is in here?' By using the weight per bag (molar mass) and number of bags (moles we calculated earlier), we can answer that question and understand how much helium we have in terms that make sense to us, whether for party decorations or scientific experiments.