Question

In the electrolysis of a sodium chloride solution, what volume of ${\mathrm{H}}_2(g)$ is produced in the same time it takes to produce $257\mathrm{L}$ ${\mathrm{Cl}}_2(g),$ with both volumes measured at $50{.}^{\circ }\mathrm{C}$ and 2.50 atm?

Short answer

The volume of hydrogen gas ($H_2$) produced in the same time it takes to produce 257 L of chlorine gas ($C{l}_2$), with both volumes measured at 50°C and 2.50 atm, is 257 L.

Step-by-step solution

Write the balanced chemical equation for the electrolysis process

The electrolysis of sodium chloride solution results in the production of hydrogen gas, chlorine gas, and sodium hydroxide. The balanced chemical equation for this process is: 2NaCl(aq) + 2H₂O(l) -> 2NaOH(aq) + H₂(g) + Cl₂(g)

List the given information and find the stoichiometric ratio of hydrogen and chlorine gases

We know the volume of chlorine gas produced is 257 L, and the conditions are 50°C and 2.50 atm. We also note from the balanced chemical equation that the stoichiometric ratio of hydrogen gas to chlorine gas is 1:1, as indicated by the coefficients.

Convert temperature to Kelvin

To use the ideal gas law, we must convert the temperature from Celsius to Kelvin: Temperature in Kelvin = Temperature in Celsius + 273.15 T(K) = 50°C + 273.15 = 323.15 K

Use stoichiometric ratio to find the volume of hydrogen gas produced

Since the ratio of hydrogen gas to chlorine gas is 1:1, the volume of hydrogen gas produced is the same as the volume of chlorine gas produced: Volume of hydrogen gas = Volume of chlorine gas = 257 L

Use the ideal gas law to determine the number of moles for chlorine gas

We can use the ideal gas law, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature, to find the number of moles for chlorine gas: n(Cl₂) = PV / RT = (2.50 atm)(257 L) / (0.0821 L*atm/mol*K)(323.15 K) = 24.39 mol

Use the stoichiometric ratio to find the number of moles of hydrogen gas

Now that we know the number of moles of chlorine gas, we can use the stoichiometric ratio to determine the number of moles of hydrogen gas, since the ratio is 1:1: n(H₂) = n(Cl₂) = 24.39 mol

Use the ideal gas law to determine the volume of hydrogen gas

Finally, we use the ideal gas law again to find the volume of hydrogen gas, now that we have the number of moles: V(H₂) = n(H₂)RT / P = (24.39 mol)(0.0821 L*atm/mol*K)(323.15 K) / (2.50 atm) = 257 L Therefore, the volume of hydrogen gas produced in the same time it takes to produce 257 L of chlorine gas, with both volumes measured at 50°C and 2.50 atm, is 257 L.

Key concepts

Stoichiometry

Stoichiometry is the concept of relating the quantities of reactants and products in a chemical reaction. It is a fundamental idea in chemistry that allows us to understand how much of each reactant is needed to produce a desired amount of product, or conversely, how much product is formed from a given amount of reactant. To do this, stoichiometry uses balanced chemical equations, which indicate the relative quantities of substances involved in the reaction. In our example with sodium chloride electrolysis, the balanced chemical equation is:$$2\text{NaCl(aq)}+2{\text{H}}_2\text{O(l)}\to 2\text{NaOH(aq)}+{\text{H}}_2\text{(g)}+{\text{Cl}}_2\text{(g)}$$This equation tells us that 2 moles of sodium chloride and 2 moles of water produce 1 mole of hydrogen gas along with 1 mole of chlorine gas and 2 moles of sodium hydroxide. The coefficients before each substance show the stoichiometric ratio.

• In this process, for every 1 mole of ${\text{Cl}}_2$ produced, 1 mole of ${\text{H}}_2$ is also produced.
• This 1:1 ratio is crucial for solving volumes and amounts in reactions.

Understanding stoichiometry lets us use these ratios to predict how much product will be made, or how much reactant is needed, under given conditions.

Ideal Gas Law

The ideal gas law is a very useful equation in chemistry that relates the pressure, volume, temperature, and amount of moles of a gas. It is represented as $PV=nRT$, where:

• $P$ represents pressure.
• $V$ is volume.
• $n$ stands for the number of moles.
• $R$ is the ideal gas constant $(0.0821\text{ L}\text{atm/mol}\text{K})$.
• $T$ is temperature in Kelvin.

In the problem of sodium chloride electrolysis, the ideal gas law helps us find the number of moles of chlorine and hydrogen gases produced. We manipulated this formula to solve for the number of moles $n$:$$n=\frac{PV}{RT}$$For the chlorine gas ${\text{Cl}}_2$, we calculated the moles using the given conditions (pressure 2.50 atm, volume 257 L, and temperature 323.15 K).Once we have the moles, we can use the stoichiometric ratio to find the moles of hydrogen gas since it has a 1:1 ratio with chlorine. Finally, we used the ideal gas law again to find the volume of hydrogen gas produced, ensuring that the solution satisfies the given conditions by converting back if necessary.

Chemical Equations

Chemical equations are symbolic representations of chemical reactions. They show substances involved as reactants transforming into products. The balanced equations are essential because they obey the law of conservation of mass, meaning matter cannot be created or destroyed. In the sodium chloride electrolysis, the balanced chemical equation is:$$2\text{NaCl(aq)}+2{\text{H}}_2\text{O(l)}\to 2\text{NaOH(aq)}+{\text{H}}_2\text{(g)}+{\text{Cl}}_2\text{(g)}$$This equation demonstrates a few important principles:

• Reactants and Products: The starting substances (on the left) are called reactants and the substances formed (on the right) are products.
• Balancing the Equation: It is crucial that the same number of each type of atom exists on both sides of the equation. For example, 2 sodium and 2 chlorine atoms are in $\text{NaCl}$, balanced by 2 sodium in $\text{NaOH}$ and 2 chlorine in ${\text{Cl}}_2$.
• State Symbols: They tell us the physical state of each substance: (aq) for aqueous, (l) for liquid, and (g) for gas.

Chemical equations provide insights into the quantitative aspects of chemical reactions, enabling us to calculate how much of each reactant is needed to produce a desired amount of product.