Question

the heat of reaction (qₓₙ) is determined to be -225 j when 0.10 mol of solute dissolves. what is the heat of solution (qₛₒₗₙ) for this substance? +225 j +2250 j -225 j -2250 j

Explanation:

Step1: Recall the relationship between \( q_{\text{rxn}} \) and \( q_{\text{soln}} \)

When a solute dissolves, the heat of the reaction (\( q_{\text{rxn}} \)) and the heat of the solution (\( q_{\text{soln}} \)) are related by \( q_{\text{soln}} = -q_{\text{rxn}} \) (based on the principle of energy conservation, the heat lost by the reaction is gained by the solution and vice versa, but here we first consider the sign relationship for the amount per mole? Wait, no, wait. Wait, actually, the heat of solution is the heat associated with dissolving the solute. But also, we need to consider the moles? Wait, no, wait the question: Wait, the heat of reaction is -225 J when 0.10 mol dissolves. Wait, no, maybe I misread. Wait, the heat of solution is the heat per mole? Wait, no, the question is: What is the heat of solution (\( q_{\text{soln}} \)) for this substance? Wait, maybe the heat of solution here is the total heat for the dissolution of 0.10 mol? No, that can't be. Wait, no, the key is the relationship between \( q_{\text{rxn}} \) and \( q_{\text{soln}} \). The reaction here is the dissolution, so \( q_{\text{rxn}} \) (the heat of the dissolution reaction) and \( q_{\text{soln}} \) (the heat gained or lost by the solution) are related by \( q_{\text{soln}} = -q_{\text{rxn}} \) when considering the system and surroundings. But also, we need to find the heat of solution per mole? Wait, no, the question is: when 0.10 mol dissolves, \( q_{\text{rxn}} = -225 \) J. Wait, maybe the heat of solution is the heat per mole, so we need to calculate the heat per mole. Wait, let's re-express:

The heat of reaction (dissolution) for 0.10 mol is \( q_{\text{rxn}} = -225 \) J. The heat of solution (\( \Delta H_{\text{soln}} \)) is the heat per mole of solute dissolved. But also, the heat of the solution (the solution's heat change) is \( q_{\text{soln}} = -q_{\text{rxn}} \) for the process, but if we want the heat of solution (the enthalpy of solution), we need to find the heat per mole. Wait, no, maybe the question is using \( q_{\text{soln}} \) as the total heat for the solution, but that doesn't make sense. Wait, no, perhaps I made a mistake. Wait, let's check the options. The options are +225 J, +2250 J, -225 J, -2250 J. So let's think again.

Wait, the heat of reaction is -225 J when 0.10 mol dissolves. So the heat of solution (the heat per mole) would be \( \frac{-q_{\text{rxn}}}{\text{moles}} \)? Wait, no. Wait, the heat of solution is the heat associated with dissolving the solute. The reaction (dissolution) releases or absorbs heat. If \( q_{\text{rxn}} = -225 \) J for 0.10 mol, that means the reaction (dissolution) is exothermic (since \( q_{\text{rxn}} \) is negative, meaning the system loses heat). Then the solution would gain heat, so \( q_{\text{soln}} = +225 \) J for 0.10 mol? But that's not one of the options. Wait, the options include +2250 J. Ah! Wait, maybe I missed that the heat of solution is per mole, so we need to calculate the heat for 1 mole. So:

For 0.10 mol, \( q_{\text{rxn}} = -225 \) J. So for 1 mole, the heat of reaction would be \( \frac{-225 \text{ J}}{0.10 \text{ mol}} = -2250 \) J/mol. But the heat of solution is the negative of the heat of reaction (because the heat of solution is the heat absorbed or released by the solution, so if the reaction releases heat (negative \( q_{\text{rxn}} \)), the solution absorbs heat (positive \( q_{\text{soln}} \)) per mole? Wait, no. Let's recall: The enthalpy of solution (\( \Delta H_{\text{soln}} \)) is the enthalpy change when 1 mole of solute dissolves in a solvent. The relationship is \( \Delta H_{\text{soln}} = \frac{q_{\text{rxn}}}{n} \), but with sign conventions. Wait, if the dissolution reaction (system) has \( q_{\text{rxn}} = -225 \) J for 0.10 mol (exothermic, system loses heat), then the enthalpy of solution (heat per mole) would be \( \Delta H_{\text{soln}} = \frac{q_{\text{rxn}}}{n} = \frac{-225 \text{ J}}{0.10 \text{ mol}} = -2250 \) J/mol. But the heat of solution (\( q_{\text{soln}} \)) for the solution (surroundings) would be \( -q_{\text{rxn}} \) per mole? Wait, no, the solution's heat change is \( q_{\text{soln}} = -q_{\text{rxn}} \) for the process. But if we want the heat of solution (the enthalpy change for the solution process), which is the heat per mole, then we need to calculate the heat per mole.

Wait, let's do the math:

Given:
- Moles of solute, \( n = 0.10 \) mol
- Heat of reaction (dissolution) for \( n = 0.10 \) mol: \( q_{\text{rxn}} = -225 \) J

The heat of solution (\( \Delta H_{\text{soln}} \)) is the heat per mole of solute dissolved. The relationship between \( q_{\text{rxn}} \) and \( \Delta H_{\text{soln}} \) is \( q_{\text{rxn}} = n \times \Delta H_{\text{soln}} \) (assuming constant pressure, \( q_p = \Delta H \)).

But also, the heat of the solution (the solution's heat change) is \( q_{\text{soln}} = -q_{\text{rxn}} \) (because the system (reaction) and surroundings (solution) exchange heat: \( q_{\text{system}} + q_{\text{surroundings}} = 0 \), so \( q_{\text{soln}} = -q_{\text{rxn}} \)).

But the question is asking for \( q_{\text{soln}} \) for this substance. Wait, maybe the question is actually asking for the heat of solution per mole, but the options are in J, not J/mol. Wait, the options are +225 J, +2250 J, -225 J, -2250 J. Let's check the sign first. If \( q_{\text{rxn}} = -225 \) J (reaction releases heat, so the solution absorbs heat, so \( q_{\text{soln}} \) should be positive). Now, is it per mole or total? Wait, the reaction is for 0.10 mol, so if we want the heat of solution (the total heat for the solution when 0.10 mol dissolves), then \( q_{\text{soln}} = -q_{\text{rxn}} = 225 \) J. But that's one of the options (+225 J), but wait, no, maybe I'm missing the moles. Wait, no, the heat of solution is the heat per mole, so we need to calculate the heat per mole. Let's see:

\( \Delta H_{\text{soln}} = \frac{q_{\text{rxn}}}{n} \) (but with sign). Wait, \( q_{\text{rxn}} = -225 \) J for 0.10 mol, so \( \Delta H_{\text{soln}} = \frac{-225 \text{ J}}{0.10 \text{ mol}} = -2250 \) J/mol. But the heat of the solution (the solution's heat change) is \( q_{\text{soln}} = -q_{\text{rxn}} = 225 \) J for 0.10 mol? No, that can't be. Wait, I think I made a mistake. Let's re-express:

The dissolution process: the system is the solute dissolving, the surroundings is the solution. The heat lost by the system (reaction) is gained by the surroundings (solution), so \( q_{\text{system}} = q_{\text{rxn}} = -225 \) J (system loses heat), so \( q_{\text{surroundings}} = q_{\text{soln}} = +225 \) J. But this is for 0.10 mol. However, the heat of solution is typically expressed per mole of solute. So to find the heat of solution (per mole), we calculate:

\( \Delta H_{\text{soln}} = \frac{q_{\text{soln}}}{n} \) (since \( q_{\text{soln}} \) is the heat gained by the solution, and the heat of solution is the enthalpy change for dissolving 1 mole, which is equal to \( \frac{q_{\text{soln}}}{n} \) if we consider the solution's heat change). Wait, no, the enthalpy of solution is \( \Delta H_{\text{soln}} = q_{\text{rxn}} / n \) (because \( q_{\text{rxn}} \) is the heat of the reaction, which is the enthalpy change for the reaction, so per mole). So \( \Delta H_{\text{soln}} = \frac{-225 \text{ J}}{0.10 \text{ mol}} = -2250 \) J/mol. But the heat of the solution (the solution's heat change) is \( q_{\text{soln}} = -q_{\text{rxn}} = 225 \) J for 0.10 mol. But the options include +2250 J. Wait, maybe the question is asking for the heat of solution per mole, so we need to calculate the heat per mole. Let's check the options:

If \( q_{\text{rxn}} = -225 \) J for 0.10 mol, then the heat of solution (per mole) is \( \frac{-q_{\text{rxn}}}{n} \)? Wait, no. Wait, the heat of solution is the heat absorbed or released when 1 mole of solute dissolves. So if 0.10 mol releases 225 J (since \( q_{\text{rxn}} = -225 \) J, meaning the reaction is exothermic, releases heat), then 1 mole would release \( \frac{225 \text{ J}}{0.10 \text{ mol}} = 2250 \) J, so the heat of solution (which is the enthalpy change, so if it's exothermic, \( \Delta H_{\text{soln}} = -2250 \) J/mol, but the heat gained by the solution would be +2250 J/mol. Wait, now I'm confused. Let's look at the options:

The options are +225 J, +2250 J, -225 J, -2250 J.

Let's think again:

The heat of reaction (dissolution) for 0.10 mol is \( q_{\text{rxn}} = -225 \) J (this means the reaction releases 225 J, so the solution absorbs 225 J for 0.10 mol). But the heat of solution is the heat per mole, so we need to find the heat absorbed by the solution per mole. So:

Heat absorbed by solution for 0.10 mol: \( q_{\text{soln}} = -q_{\text{rxn}} = 225 \) J.

Heat absorbed by solution per mole: \( \frac{225 \text{ J}}{0.10 \text{ mol}} = 2250 \) J/mol.

So the heat of solution (the enthalpy change for the solution, which is the heat gained by the solution per mole) is +2250 J/mol. Therefore, the answer is +2250 J.

Wait, that makes sense. Because when 0.10 mol dissolves, the reaction releases 225 J (so \( q_{\text{rxn}} = -225 \) J), so the solution gains 225 J. To find the heat per mole, we divide by 0.10 mol, giving 2250 J/mol. So the heat of solution is +2250 J (per mole, but the question says "heat of solution (q_soln) for this substance"—maybe it's the total heat for 1 mole? Or the heat per mole. The options include +2250 J, so that must be it.

Step2: Calculate the heat of solution

Given \( q_{\text{rxn}} = -225 \) J for \( n = 0.10 \) mol.

The heat of solution (\( q_{\text{soln}} \)) per mole is calculated by considering the relationship between the heat of the reaction and the moles of solute. Since the heat gained by the solution is the negative of the heat lost by the reaction, and we need the heat per mole:

First, find the heat per mole of solute dissolved. The heat of reaction for 0.10 mol is -225 J, so the heat per mole is:

\( \frac{-q_{\text{rxn}}}{n} = \frac{-(-225 \text{ J})}{0.10 \text{ mol}} = \frac{225 \text{ J}}{0.10 \text{ mol}} = 2250 \text{ J/mol} \)

Since the solution gains heat (the dissolution reaction is exothermic, so the solution absorbs heat), the heat of solution is positive, and the value is 2250 J (per mole, but the question's options are in J, so likely referring to the heat per mole or the total heat for 1 mole? Wait, the question says "heat of solution (q_soln) for this substance"—maybe it's the heat per mole, so the answer is +2250 J.

Answer:

+2250 J