Problem 79 - Chemistry: An Atoms-Focused Approach

Question

Do all gases at the same pressure and temperature have the same density? Explain your answer.

Step-by-Step Solution

Answer

Explain your answer. Answer: No, not all gases at the same pressure and temperature have the same density. The density of a gas is given by the formula ρ = PM/(RT), where P is pressure, M is molar mass, R is the gas constant, and T is temperature. Different gases have different molar masses, which ultimately affects their densities even when they have the same pressure and temperature.

Step 1: Recall the Ideal Gas Equation

The ideal gas equation is given by: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.

Step 2: Density Formula

We know that density (ρ) is defined as mass (m) per unit volume (V). The formula for density is: ρ = m/V

Step 3: Relating Ideal Gas Equation with Density

We can rewrite the ideal gas equation in terms of mass (m) and molar mass (M) instead of the number of moles (n). This helps us relate it to the density formula. Knowing that n = m/M, we can replace n in the ideal gas equation: PV = (m/M)RT Now, we can isolate the mass-to-volume ratio (m/V), which is the density (ρ): ρ = PM/(RT)

Step 4: Comparing Densities

Now we can see that the density of a gas is given by the formula ρ = PM/(RT). If two different gases are at the same pressure (P) and temperature (T), we need to look at their molar masses (M) to determine their densities.

Step 5: Conclusion

In conclusion, not all gases at the same pressure and temperature have the same density because their densities are affected by their molar masses (M). Different gases have different molar masses, which ultimately affects their densities even when they have the same pressure and temperature.