Question

Hydrogen gives off \(120 . \mathrm{J} / \mathrm{g}\) of energy when burned in oxygen, and methane gives off \(50 . \mathrm{J} / \mathrm{g}\) under the same circumstances. If a mixture of 5.0 \(\mathrm{g}\) hydrogen and \(10 . \mathrm{g}\) methane is burned, and the heat released is transferred to 50.0 \(\mathrm{g}\) water at \(25.0^{\circ} \mathrm{C},\) what final temperature will be reached by the water?

Short answer

The final temperature of the water will be approximately \(30.24 ^{\circ}\mathrm{C}\).

Step-by-step solution

Calculate the energy released by burning hydrogen

First, we need to find out how much energy is released when 5.0 g of hydrogen is burned. We are given that hydrogen releases 120 J/g when burned in oxygen. Therefore, we can multiply the mass of hydrogen by the energy released per gram to find the total energy released by the hydrogen. Total energy released by hydrogen = (mass of hydrogen) × (energy released per gram) = 5.0 g × 120 J/g = 600 J

Calculate the energy released by burning methane

Next, we need to find out how much energy is released when 10 g of methane is burned. We are given that methane releases 50 J/g when burned in oxygen. Therefore, we can multiply the mass of methane by the energy released per gram to find the total energy released by the methane. Total energy released by methane = (mass of methane) × (energy released per gram) = 10 g × 50 J/g = 500 J

Calculate the total energy released

Add the energy released by burning hydrogen and methane to find the total energy released. Total energy released = energy from hydrogen + energy from methane = 600 J + 500 J = 1100 J

Calculate the temperature change of the water

Now, we'll use the energy released to find the temperature change in the 50.0 g of water. We will use the formula Q = mcΔT, where Q is the heat absorbed (in joules), m is the mass of the substance absorbing the heat (in grams), c is the specific heat capacity of the substance (in J/g°C), and ΔT is the temperature change (in °C). For water, the specific heat capacity is approximately 4.18 J/g°C. We are given the mass of water (50.0 g) and the energy we calculated in Step 3 (1100 J). We can rearrange the formula to solve for ΔT: ΔT = Q / (mc) Temperature change in water = total energy released / (mass of water × specific heat of water) = 1100 J / (50.0 g × 4.18 J/g°C) ≈ 5.24 °C

Calculate the final temperature of the water

Now that we have the temperature change in the water, we can add this value to the initial temperature of the water (25.0 °C) to find the final temperature: Final temperature of water = initial temperature + temperature change = 25.0 °C + 5.24 °C ≈ 30.24 °C So the final temperature of the water will be approximately \(30.24 ^{\circ}\mathrm{C}\).

Key concepts

Heat Transfer

Heat transfer is a fascinating concept that describes the movement of energy between substances due to a difference in temperature. When we burn substances like hydrogen and methane, they release energy in the form of heat. This heat then transfers to another substance---in our case, water.

In calorimetry problems, we often deal with heat transfer from a chemical reaction to a surrounding substance, often resulting in a change in temperature. This process consists of:

• Exothermic reactions that give off heat (like our burning fuel),
• Endothermic reactions that absorb heat.

Heat transfer is directional; it always moves from the hotter object to a cooler one. This fundamental principle helps us predict resultant changes in temperature and energy distribution between substances.

Specific Heat Capacity

Specific heat capacity is a property that describes how much energy is required to raise the temperature of one gram of a substance by one degree Celsius. Different materials require varying amounts of energy to change their temperature.

For water, this amount is notably high, about 4.18 J/g°C. This means that water can absorb or release a lot of energy with only a small change in temperature. In our exercise, understanding water's specific heat capacity allows us to determine how much its temperature will change when it absorbs the heat released from burning fuels.

Calculating temperature changes with specific heat capacity involves the formula:

• \[ Q = mc\Delta T \]
• Here: \( Q \) is heat energy, \( m \) is mass, \( c \) is specific heat capacity, and \( \Delta T \) is the temperature change.

This ability to calculate changes is key to predicting outcomes in chemical reactions and heat transfer scenarios.

Enthalpy of Combustion

The enthalpy of combustion is the energy released as heat when a substance combusts in adequate oxygen conditions. It's usually expressed in Joules per gram (J/g) or kilojoules per mole (kJ/mol).

In our given problem, the enthalpy of combustion is provided for both hydrogen and methane:

• Hydrogen releases 120 J/g,
• Methane releases 50 J/g.

Understanding this property allows us to calculate how much energy the burning of these fuels releases. By knowing the mass of each fuel, we compute the total energy that affects other substances like water. This data is crucial to solving calorimetry and heat transfer problems effectively.

Temperature Change

Temperature change is a critical outcome of heat transfer. It's what we ultimately seek to understand in calorimetry problems: how does the temperature of a substance like water change as it absorbs or expels heat?

Using the formula \( \Delta T = \frac{Q}{mc} \), we translate the total energy absorbed (1100 J in our problem) and specific heat capacity to find out how much the temperature of water changes. Calculations showed:

• Initial water temperature: 25.0°C,
• Energy absorbed: 1100 J,
• Resulting Temperature Change: ≈ 5.24 °C.

Therefore, the final temperature was calculated to be around 30.24°C. This kind of precise change depends on a precise understanding of thermodynamic principles and careful computational steps.