Chapter 12: Problem 88
Using the ideal gas law, \(P V=n R T\), calculate the following: (a) the volume of \(0.510 \mathrm{~mol}\) of \(\mathrm{H}_{2}\) at \(47^{\circ}\mathrm{C}\) and \(1.6\) atm pressure (b) the number of grams in \(16.0 \mathrm{~L}\) of \(\mathrm{CH}_{4}\) at\(27^{\circ} \mathrm{C}\) and 600 . torr pressure (c) the density of \(\mathrm{CO}_{2}\) at \(4.00\) atm pressure and \(20.0^{\circ}\mathrm{C}\) (d) the molar mass of a gas having a density of \(2.58 \mathrm{~g} /\mathrm{L}\) at \(27^{\circ} \mathrm{C}\) and \(1.00\) atm pressure.
Short Answer
Expert verified
(a) Volume of H2 = 10.4 L. (b) Mass of CH4 = 8.07 g. (c) Density of CO2 = 1.80 g/L. (d) Molar mass of gas = 65.2 g/mol.
Step by step solution
01
Convert the temperature to Kelvin
To use the ideal gas law, we need the absolute temperature in Kelvin. Convert the Celsius temperature to Kelvin by adding 273.15.(a) For hydrogen, the temperature is: T = 47°C + 273.15 = 320.15 K(b) For methane, the temperature is: T = 27°C + 273.15 = 300.15 K(c) For carbon dioxide, the temperature is: T = 20.0°C + 273.15 = 293.15 K(d) For the unknown gas, the temperature is: T = 27°C + 273.15 = 300.15 K
02
Using the ideal gas equation for volume
(a) Insert the known values into the ideal gas law and solve for the volume.Given: n = 0.510 mol, P = 1.6 atm, T = 320.15 K, R = 0.0821 L*atm/(mol*K).The ideal gas law is: P x V = n x R x TRearrange to solve for V: V = (n x R x T) / PSubstitute the values into the equation: V = (0.510 mol x 0.0821 L*atm/(mol*K) x 320.15 K) / 1.6 atm
03
Find the number of moles of methane
(b) Insert the known values into the ideal gas law and solve for the number of moles (n).Given: V = 16.0 L, P = 600 torr (1 atm = 760 torr), T = 300.15 K.First, convert pressure from torr to atm: P = 600 torr * (1 atm / 760 torr).Use the ideal gas law, rearranged to solve for n: n = (P x V) / (R x T).Substitute the values into the formula and solve for n.
04
Calculate the mass of methane
(b) Given the molar mass of methane (CH4) is 16.04 g/mol, use the number of moles calculated in Step 3 to find the mass.Mass = n x molar mass of CH4.
05
Solving for the density of CO2
(c) Density is mass per unit volume. Using the ideal gas law, the density can be expressed as density = (mass / V) = (n x M) / V, where M is the molar mass of CO2.Since P x V = n x R x T, we can derive that n = (P x V) / (R x T).Thus, Density = P x M / (R x T).Substitute the values for P, M, R, T into the equation to find the density of CO2.
06
Calculate the molar mass of the unknown gas
(d) Given the density (d), we can rearrange the ideal gas law to solve for the molar mass (M).The formula for density in terms of the ideal gas law is d = (M x P) / (R x T).Rearrange to solve for M: M = d x R x T / P.Substitute the known values for d, R, T, and P into the equation to calculate M.
07
Perform the necessary calculations
(a) Calculate the volume of hydrogen.(b) Calculate the number of moles and mass of methane.(c) Calculate the density of carbon dioxide.(d) Calculate the molar mass of the unknown gas.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gas Volume Calculation
Understanding how to calculate the volume of a gas is a fundamental concept in chemistry, particularly when dealing with reactions and processes involving gases. The volume of a gas can be predicted using the ideal gas law, expressed as the formula: \(PV = nRT\).
To calculate the gas volume, this relationship requires us to know the number of moles of the gas (\(n\)), the temperature (\(T\)) in Kelvin, the pressure (\(P\)) exerted by the gas, and the ideal gas constant (\(R\)). For example, in the given exercise where we have 0.510 moles of hydrogen gas (\(H_2\)) at a temperature of 320.15 K and a pressure of 1.6 atm, the volume (\(V\)) can be calculated by rearranging the formula to solve for \(V\): \(V = \frac{nRT}{P}\).
Once you plug the known values into the equation, the calculation yields the volume of the hydrogen gas under the specified conditions. This method is widely used in laboratory settings and industrial applications to determine the required volume of a gas for a particular reaction or process. It is also a stepping stone toward more complex gas-related calculations.
Molar Mass Determination
The molar mass of a substance is the mass of one mole of that substance, usually expressed in grams per mole (g/mol). It plays a crucial role in chemistry when converting between the mass of a substance and the number of moles. For gases, the ideal gas law provides a handy way to determine molar mass if other properties of the gas are known.
In the context of the ideal gas law, the molar mass (\(M\)) can be calculated by rearranging the density form of the law: \(d = \frac{MP}{RT}\), which gives us \(M = \frac{dRT}{P}\). Here, density (\(d\)) is mass per unit volume. This concept is beautifully illustrated in part (d) of the exercise, where the unknown gas has a density of 2.58 g/L at a temperature of 300.15 K and a pressure of 1.00 atm. By rearranging the equation and substituting the known values, we can solve for the molar mass of the gas.
To illustrate, imagine a balloon filled with an unknown gas. By measuring the mass and volume of the gas in the balloon, while knowing the temperature and pressure, one can calculate the molar mass and potentially identify the gas. Molar mass determination is thus essential for chemical identification and reaction stoichiometry.
Gas Density Computation
Gas density is a measure of a gas's mass per unit volume and is an important property in both scientific and industrial applications. Understanding how to compute the gas density involves manipulating the ideal gas law equation appropriately.
Starting with the ideal gas law, \(PV = nRT\), and knowing that the number of moles (\(n\)) is equal to mass (\(m\)) divided by the molar mass (\(M\)), we can rewrite the equation in terms of mass and volume. This gives us the density equation: \(d = \frac{PM}{RT}\), where \(d\) is density, \(P\) is pressure, \(M\) is molar mass, \(R\) is the ideal gas constant, and \(T\) is temperature. Knowing any three of these properties allows us to calculate the fourth.
For instance, as performed in part (c) of the exercise, the density of carbon dioxide (\(CO_2\)) at 4.00 atm pressure and 293.15 K temperature can be computed by substituting the known values into the density equation. In practical scenarios, this computation enables us to design appropriate storage systems for gases and helps in determining the buoyancy and lifting properties of gases, which is vital for applications like hot air balloons and even predicting the atmosphere's behavior.
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