Question

A student prepared \(118.9 \mathrm{~mL}\) of \(\mathrm{CO}_{2}\) at a pressure of \(758 \mathrm{~mm} \mathrm{Hg}\) and a temperature of \(22^{\circ} \mathrm{C}\). He did this by adding \(35.47 \mathrm{~mL}\) of a \(0.1380 \mathrm{M}\) solution of a strong acid \(\left(\mathrm{H}^{+}\right)\) to \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) $$ \mathrm{Na}_{2} \mathrm{CO}_{3}(s)+2 \mathrm{H}^{+}(a q) \longrightarrow 2 \mathrm{Na}^{+}(a q)+\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O} $$ Did the student use \(\mathrm{HCI}\) or \(\mathrm{H}_{2} \mathrm{SO}_{4}\) as the strong acid?

Short answer

Answer: The strong acid used in the reaction with Na2CO3 was H2SO4.

Step-by-step solution

Calculate the number of moles of CO2 gas produced

First, we need to convert the given temperature from Celsius to Kelvin: $$T(K)=T(°C)+273.15$$ $$T(K)=22+273.15=295.15 K$$ Now, use the Ideal Gas Law (PV=nRT) to calculate the number of moles of CO2 produced. We need to convert the pressure from mm Hg to atm by dividing it by 760 mmHg/atm: $$n=\dfrac{PV}{RT}$$ $$n=\dfrac{(758 \mathrm{~mm} \mathrm{Hg})(118.9 \mathrm{~mL})}{(62.36 \mathrm{L}\, \mathrm{mm} \mathrm{Hg} /\mathrm{mol} \, \mathrm{K})(295.15 \mathrm{~K})}$$ $$n=\dfrac{(758/760 \mathrm{~atm})(118.9/1000 \mathrm{~L})}{0.0821 \mathrm{L}\, \mathrm{atm} /(\mathrm{mol} \, \mathrm{K}) (295.15 \mathrm{~K})}$$ After the calculations, we have: $$n \approx 0.0050 \mathrm{~mol} \, \mathrm{CO}_{2}$$

Determine moles of H+ ions required

Using the stoichiometry of the reaction, we can determine that 2 moles of H+ ions are required to produce 1 mole of CO2: $$ \mathrm{Na}_{2} \mathrm{CO}_{3}(s)+2 \mathrm{H}^{+}(a q) \longrightarrow 2 \mathrm{Na}^{+}(a q)+\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}$$ So, we can calculate the moles of H+ ions as: $$\text{moles of }\mathrm{H}^{+} = 2\times \text{moles of }\mathrm{CO}_{2}$$ $$\text{moles of }\mathrm{H}^{+} = 2\times 0.0050 \mathrm{~mol} \, \mathrm{CO}_{2} = 0.0100 \mathrm{~mol}\, \mathrm{H}^{+}$$

Calculate the number of moles of H+ ions in the strong acid used

We are given the volume and molarity of the strong acid used, so we can calculate the number of moles of H+ ions: $$\text{moles of }\mathrm{H}^{+} = (\text{volume of strong acid})(\text{molarity of strong acid})$$ $$\text{moles of }\mathrm{H}^{+} = (35.47\, \mathrm{mL})(0.1380 \mathrm{M})=0.00489 \mathrm{~mol}\, \mathrm{H}^{+}$$

Identify the strong acid

Now we will use the ratio of moles of H+ ions in the strong acid to the moles of H+ ions needed to produce the measured amount of CO2 to identify the strong acid: $$\text{ratio of }\mathrm{H}^{+}\text{ ions} = \dfrac{\text{moles of }\mathrm{H}^{+}\text{ in strong acid}}{\text{moles of }\mathrm{H}^{+}\text{ needed}}$$ $$\text{ratio of }\mathrm{H}^{+}\text{ ions} = \dfrac{0.00489\, \mathrm{mol}\, \mathrm{H}^{+}}{0.0100\, \mathrm{mol}\,\mathrm{H}^{+}} \approx 0.49$$ If the strong acid were HCl, the ratio should be close to 1, because there is one mole of H+ ions in HCl. If the strong acid were H2SO4, the ratio should be close to 0.5, because there are 2 moles of H+ ions in H2SO4. In our case, the ratio of H+ ions is close to 0.49 (approximately equal to 0.50), which suggests that the strong acid was \(\mathrm{H}_{2}\mathrm{SO}_{4}\).