Question

Hydrogen gives off \(120 .\) J/g of energy when burned in oxygen, and methane gives off \(50 .\) J/g under the same circumstances. If a mixture of 5.0 g hydrogen and \(10 .\) 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 water's final temperature will be approximately 30.26°C when the heat released from the mixture of 5.0 g hydrogen and 10.0 g methane is transferred to it.

Step-by-step solution

Calculate the energy released by hydrogen and methane

We need to find the total energy released by the hydrogen and methane when they burn. To do this, we will multiply the energy per gram by the total mass of each substance. Energy released by hydrogen = mass of hydrogen × energy per gram of hydrogen Energy released by hydrogen = 5.0 g × 120 J/g = 600 J Energy released by methane = mass of methane × energy per gram of methane Energy released by methane = 10.0 g × 50 J/g = 500 J

Calculate the total energy released by the mixture

Now that we have the energy released by hydrogen and methane separately, we will add them together to get the total energy released by the mixture. Total energy released = Energy released by hydrogen + Energy released by methane Total energy released = 600 J + 500 J = 1100 J

Use the equation for heat transfer to determine the final temperature of the water

We will use the heat transfer equation: Q = mcΔT, where Q is the energy released or absorbed, m is the mass of the substance, c is the specific heat capacity of the substance (for water, it is 4.18 J/g°C), and ΔT is the change in temperature. First, let's rearrange the equation to solve for ΔT: ΔT = Q / (mc) Now plug in the values we know: ΔT = 1100 J / (50.0 g × 4.18 J/g°C) ΔT ≈ 5.26°C Since we want to find the final temperature, we will add the initial temperature to the change in temperature: Final temperature = Initial temperature + ΔT Final temperature = 25.0°C + 5.26°C Final temperature ≈ 30.26°C The water's final temperature will be approximately 30.26°C when the heat released from the mixture is transferred to it.

Key concepts

Energy Released by Combustion

Combustion is a chemical process in which a substance combines with oxygen to release energy, predominantly in the form of heat. The energy released during combustion can be quantified as the amount of energy produced per gram of the burning material. This value is critical for solving problems related to heat transfer in chemical reactions.

For instance, in the exercise provided, hydrogen releases 120 Joules per gram, and methane releases 50 Joules per gram when they combust. By multiplying these values by the respective masses of hydrogen and methane, we can determine the total amount of energy released. Understanding the specific energy values for various fuels allows us to predict the extent of the temperature increase in a substance, like water, that absorbs the heat.

Specific Heat Capacity

Specific heat capacity is a property of a substance that indicates how much energy is needed to raise the temperature of one gram of the substance by one degree Celsius. It's measured in joules per gram per degree Celsius (\( J/g^{\text{o}}C \)).

Role in Heat Transfer

Water, for example, has a high specific heat capacity of 4.18 J/g°C. This means it can absorb a considerable amount of heat without a significant rise in temperature. This is why, despite receiving 1100 J of energy in our exercise, the water's temperature only increases modestly. This property plays a crucial role in calculating how much a temperature will rise when a certain amount of energy is introduced.

Temperature Change Calculation

Calculating the temperature change of a substance after gaining or losing heat involves understanding both the energy involved and the specific heat capacity of the substance. The actual change in temperature (\( \triangle T \) is found by taking the total energy transferred and dividing it by the product of the mass of the substance and its specific heat capacity.

Practical Application

In practical scenarios such as cooking or heating systems, knowing how to calculate the change in temperature helps control processes. Applying this to our exercise, once the total heat energy transferred to the water is determined, we use the specific heat capacity and the mass of the water to find out how much the water's temperature will rise.

Heat Transfer Equation

The heat transfer equation, \( Q = mc\triangle T \) connects the energy transferred, the mass of the substance absorbing or releasing heat, its specific heat capacity, and the change in temperature. The symbol \( Q \) stands for the energy in joules, \( m \) for mass in grams, \( c \) for the specific heat capacity, and \( \triangle T \) for the change in temperature in degrees Celsius.

Understanding the Formula

It's useful to rearrange the formula when solving for different variables. For instance, when we have the amount of energy and need to calculate the final temperature, as the exercise demonstrates. This formula's versatility makes it indispensable in industries where temperature regulation is critical, such as food manufacturing or metallurgical processes.