Steve, in an attempt to impress his girlfriend, plops 18.2 grams of sodium metal into 500 mL of concentrated phosphoric acid (density = 1.88 g/cm3) in a beaker. The reactants were both initially at 25 oC. The reaction gently produced bubbles of hydrogen and Janet was not impressed. What was the final temperature of the reaction mixture if all the heat is transferred to/from the phosphoric acid?

Question

Steve, in an attempt to impress his girlfriend, plops 18.2 grams of sodium metal into 500 mL of concentrated phosphoric acid (density = 1.88 g/cm3) in a beaker. The reactants were both initially at 25 oC. The reaction gently produced bubbles of hydrogen and Janet was not impressed. What was the final temperature of the reaction mixture if all the heat is transferred to/from the phosphoric acid?

2 H3PO4 (l) + 6 Na (s) = 2 Na3PO4 (s) + 3 H2 (g)
Thermodynamic data for phosphoric acid can be found on the National Institutes of Standards and Technology website:
webbook.nist.gov/cgi/cbook.cgi?ID=C7664382&Mask=2&Type=JANAFL&Table=on#JANAFL).

Answer 1

To calculate the final temperature of the reaction mixture, we need to make use of the heat transfer equation:

q = mcΔT

where:
q is the amount of heat transferred
m is the mass of the substance
c is the specific heat capacity of the substance
ΔT is the change in temperature

In this case, we want to know the final temperature, so we rearrange the equation as:

ΔT = q / (mc)

First, we will start by calculating the heat transferred (q) from the reaction of sodium metal with phosphoric acid. This can be done by using the balanced equation and stoichiometry:

2 H3PO4 (l) + 6 Na (s) → 2 Na3PO4 (s) + 3 H2 (g)

Since the reaction produces 3 moles of hydrogen gas for every 6 moles of sodium, we can determine the amount of heat transferred by knowing the moles of sodium reacted.

1 mole of sodium has a molar mass of 22.99 g/mol, so 18.2 grams of sodium would be:

moles of Na = mass / molar mass
moles of Na = 18.2 g / 22.99 g/mol
moles of Na = 0.792 mol

Now, we need to calculate the heat transferred from the reaction. We can use the enthalpy change (ΔH) for the reaction from the provided link to the NIST website.

ΔH = -1424 kJ/mol

To calculate the heat transferred (q) from the reaction, we can use the equation:

q = ΔH × moles of Na

q = -1424 kJ/mol × 0.792 mol
q ≈ -1127.808 kJ

Next, we need to determine the specific heat capacity (c) of phosphoric acid. Unfortunately, the specific heat capacity for phosphoric acid is not provided directly in the question. However, we can make use of the density (ρ) and the formula:

density = mass / volume

to determine the mass of the phosphoric acid in the beaker. Given that the density of the phosphoric acid is 1.88 g/cm^3 and the volume is 500 mL, we can convert the volume to grams:

mass of phosphoric acid = density × volume
mass of phosphoric acid = 1.88 g/cm^3 × 500 mL × (1 cm^3 / 1 mL)
mass of phosphoric acid = 940 g

Now we can use the specific heat capacity of water (since phosphoric acid is aqueous) to approximate the specific heat capacity of phosphoric acid. The specific heat capacity of water is 4.184 J/g°C.

c ≈ 4.184 J/g°C

Finally, we can substitute the values into the rearranged equation:

ΔT = q / (mc)
ΔT = -1127.808 kJ / (940 g × 4.184 J/g°C)
ΔT ≈ -285.0086°C

Since we were given that both reactants were initially at 25°C, we add the calculated ΔT to this initial temperature to find the final temperature of the reaction mixture:

final temperature = 25°C + (-285.0086°C)
final temperature ≈ -260.0086°C

The final temperature of the reaction mixture, assuming all the heat is transferred to/from the phosphoric acid, is approximately -260.0086°C.