Problem 127 - Chemistry: An Atoms-Focused Approach

Question

How many grams of liquid methanol must be combusted to raise the temperature of \(454 \mathrm{g}\) of water from \(20.0^{\circ} \mathrm{C}\) to \(50.0^{\circ} \mathrm{C} ?\) Assume that the transfer of heat to the water is \(100 \%\) efficient. How many grams of carbon dioxide are produced in this combustion reaction?

Step-by-Step Solution

Answer

Answer: To raise the temperature of 454 grams of water from 20.0°C to 50.0°C, 2.515 grams of liquid methanol are needed. In this process, 3.450 grams of carbon dioxide will be produced.

Step 1: Calculate the heat needed to raise the temperature of water

To find out how much heat is needed to raise the temperature of the given amount of water, we will use the formula: \(Q = m \times c \times \Delta T\) where \(Q\) is the heat needed, \(m\) is the mass of the water, \(c\) is the specific heat capacity of water, and \(\Delta T\) is the change in temperature. For water, \(c = 4.18 \mathrm{\frac{J}{g \cdot °C}}\). The mass of water is given as \(454 \mathrm{g}\). The initial temperature is \(20.0^{\circ} \mathrm{C}\) and the final temperature is \(50.0^{\circ} \mathrm{C}\), giving us a change in temperature, \(\Delta T = 30.0^{\circ} \mathrm{C}\). Now we can calculate the heat needed.

Step 2: Calculate the heat needed for the temperature change

Using the formula from step 1, we can now calculate the amount of heat needed to raise the temperature of \(454\mathrm{g}\) of water from \(20.0^{\circ} \mathrm{C}\) to \(50.0^{\circ} \mathrm{C}\): \(Q = m \times c \times \Delta T = (454\mathrm{g}) \times (4.18 \mathrm{\frac{J}{g \cdot °C}}) \times (30.0^{\circ}\mathrm{ C}) = 57012 \mathrm{J}\) So, we need \(57012 \mathrm{J}\) of heat to raise the temperature of the given amount of water.

Step 3: Calculate the amount of methanol needed to produce the heat

Now we need to find out how many grams of methanol are required to produce \(57012 \mathrm{J}\) of heat. The heat of combustion of methanol (\(\Delta H_c\)) is approximately \(-726 \mathrm{\frac{kJ}{mol}}\). We will use stoichiometry to find the grams of methanol required: 1. Convert the heat needed into kJ: \(Q = 57.012\mathrm{ kJ}\) 2. Divide the heat needed by the heat of combustion of methanol to find the moles of methanol required: \(n = \frac{Q}{\Delta H_c} = \frac{57.012\mathrm{ kJ}}{-726\mathrm{\frac{kJ}{mol}}} = 0.0785\mathrm{ mol}\) 3. Convert moles of methanol to grams: \(m = (0.0785\mathrm{ mol}) \times (32.04\mathrm{\frac{g}{mol}}) = 2.515\mathrm{g}\) Therefore, \(2.515\mathrm{g}\) of methanol is required to heat the water.

Step 4: Calculate the amount of carbon dioxide produced

Finally, we need to find out how many grams of carbon dioxide are produced in this combustion reaction. The balanced combustion reaction for methanol is: \(2 \mathrm{CH_3OH} + 3 \mathrm{O_2} \rightarrow 2 \mathrm{CO_2} + 4 \mathrm{H_2O}\) From this balanced equation, we can see that 2 moles of methanol produce 2 moles of CO₂. Thus, we have a 1:1 mole ratio between methanol and CO₂. 1. Use the moles of methanol from step 3: \(0.0785\mathrm{ mol}\) 2. Calculate the moles of CO₂ produced: \(0.0785\mathrm{ mol} \times \frac{1 \mathrm{mol\, CO_2}}{1 \mathrm{mol\, CH_3OH}} = 0.0785\mathrm{ mol}\) (since the mole ratio is 1:1) 3. Convert moles of CO₂ to grams: \(m = (0.0785\mathrm{ mol}) \times (44.01\mathrm{\frac{g}{mol}}) = 3.450\mathrm{g}\) So, in this reaction, \(3.450\mathrm{g}\) of carbon dioxide will be produced. In conclusion, \(2.515\mathrm{g}\) of liquid methanol must be combusted to raise the temperature of \(454 \mathrm{g}\) of water from \(20.0^{\circ} \mathrm{C}\) to \(50.0^{\circ} \mathrm{C}\). In this process, \(3.450\mathrm{g}\) of carbon dioxide will be produced.

Key Concepts

Heat Calculation

Understanding heat calculation is essential in this exercise, as it determines how much energy is required for a specific temperature change. The heat needed, also known as calorific energy, is calculated using the formula \( Q = m \times c \times \Delta T \). Here, \( m \) is the mass of the substance being heated, \( c \) represents the specific heat capacity, and \( \Delta T \) is the change in temperature. The specific heat capacity \( c \) is a property that differs with each substance, and for water, it is typically \( 4.18 \mathrm{ \frac{J}{g \cdot °C} } \). This heat calculation helps us determine the total energy needed, a crucial step before calculating how much methanol is required.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. In the context of our exercise, it involves calculating the amount of methanol needed to supply the exact energy required to heat the water. It also aids in determining the mass of carbon dioxide produced from this combustion reaction.

• First, convert the calculated heat (in joules) into kilojoules to match the unit of methanol's heat of combustion.
• Then, determine the moles of methanol needed using the formula \( n = \frac{Q}{\Delta H_c} \), where \( \Delta H_c \) stands for the heat of combustion of methanol.
• Finally, convert these moles to grams, allowing you to calculate how much methanol must be combusted for the desired outcome.

This stoichiometric approach ensures the efficient use of methanol and aids in minimizing resource wastage.

Heat of Combustion

The heat of combustion is the amount of energy released during the combustion of a substance. In this exercise, methanol has a heat of combustion of roughly \(-726 \mathrm{ \frac{kJ}{mol} }\). This signifies how much energy is released when one mole of methanol is completely combusted. When solving the problem, this value is crucial because it helps to find out how many moles of methanol are required to raise the temperature of 454 g of water by 30°C.

• A negative sign indicates that the reaction is exothermic, meaning it releases energy.
• Using this energy value, you can work backwards to find the moles and eventually the mass of methanol needed for the task.

Understanding the heat of combustion is key in fuel-related calculations and is indispensable in optimizing fuel efficiency.

Chemical Equations

Chemical equations are representations of chemical reactions, showing the reactants and products with their respective stoichiometric coefficients. In this problem, the combustion of methanol is illustrated by the equation:

\( 2 \mathrm{CH_3OH} + 3 \mathrm{O_2} \rightarrow 2 \mathrm{CO_2} + 4 \mathrm{H_2O} \)

• This equation tells us that for every 2 moles of methanol reacted, 2 moles of carbon dioxide are produced, indicating a 1:1 molar ratio between methanol and carbon dioxide.
• By utilizing this balanced equation, you can accurately predict the amounts of all reactants and products involved. This is instrumental in the prior stoichiometry calculations.

Accurate chemical equations are fundamental in ensuring that chemical processes are well-understood and carried out efficiently.