Problem 16 - Chemistry for Allied Health

Question

98.3 J of heat is supplied to \(12.28 \mathrm{~g}\) of a substance, and its temperature rises by \(5.42^{\circ} \mathrm{C}\). What is the specific heat of the substance?

Step-by-Step Solution

Answer

The specific heat of the substance is approximately 1.474 J/g°C.

Step 1: Understanding the Concept

The specific heat capacity (c) is the amount of heat required to change the temperature of one gram of a substance by one degree Celsius. The formula to calculate specific heat is given by: \[ q = mc\Delta T \] where \( q \) is the heat added (in Joules), \( m \) is the mass (in grams), \( c \) is the specific heat (in J/g°C), and \( \Delta T \) is the change in temperature (in °C).

Step 2: Identifying Known Values

From the problem, the known values are: - Heat added, \( q = 98.3 \) J- Mass, \( m = 12.28 \) g- Change in temperature, \( \Delta T = 5.42^{\circ} \text{C} \).We need to find the specific heat \( c \).

Step 3: Rearranging the Formula

Rearrange the specific heat formula to solve for \( c \): \[ c = \frac{q}{m \Delta T} \]

Step 4: Substituting the Known Values

Substitute the known values into the rearranged formula: \[ c = \frac{98.3}{12.28 \times 5.42} \]

Step 5: Calculating the Specific Heat

Perform the calculation step by step:1. Compute \( m \Delta T = 12.28 \times 5.42 = 66.6976 \).2. Compute \( c = \frac{98.3}{66.6976} \approx 1.474 \) J/g°C.

Key Concepts

Heat Transfer

Heat transfer is the process by which thermal energy is exchanged between physical systems. It involves the movement of heat from a warmer object to a cooler one. In our exercise, 98.3 joules of heat was provided to a substance. This means that thermal energy was transferred to raise the temperature of this substance. Understanding heat transfer is crucial because it helps us explain how and why substances change temperature or phase.
There are three key mechanisms of heat transfer:

• Conduction: Transfer of heat through a medium without any movement of the medium itself. Think of a metal rod getting hot at one end when the other end is placed in a flame.
• Convection: Transfer of heat by the movement of a fluid (liquid or gas). A good example is how warm air rises and cool air sinks, circulating heat.
• Radiation: Transfer of energy through electromagnetic waves. The warmth from sunlight reaching the Earth involves radiant heat transfer.

When dealing with specific problems like ours, we focus on the total heat transferred, which is quantified by the formula: \[ q = mc\Delta T \]where \( q \) is the energy in joules. Identifying the mode of heat transfer can often help in understanding the underlying physical principles better.

Temperature Change

Temperature change reflects how much the temperature of a substance increases or decreases when heated or cooled. In this exercise, the temperature change is denoted as \( \Delta T \), which was a rise of \( 5.42^{\circ} \mathrm{C} \).
Understanding temperature change is vital because it tells us how responsive a material is to heat. This, in extension, helps us determine the specific heat capacity. Specific heat capacity is an intrinsic property indicating how much heat a specific amount of substance requires to change its temperature by one degree Celsius.
Several factors can affect the temperature change, such as:

• Type of Material: Different materials have unique specific heat capacities. For example, metals typically have lower specific heat capacities than water, meaning they heat up faster.
• Amount of Heat Added: More heat incurs a more significant temperature rise, assuming mass and material specificity remain constant.
• Initial Temperature: Changing from a colder to a warmer environment, or vice versa, requires different amounts of heat.

The formula \( q = mc\Delta T \) allows us to connect these observations and quantify the relationship between heat added, mass, and the resulting temperature change.

Mass

Mass is a measure of the amount of matter in an object, typically expressed in grams or kilograms. In our scenario, the mass of the substance being heated was \( 12.28 \) grams. Mass directly influences the capacity of a substance to absorb heat and resultant temperature change.
When we talk about heat transfer in relation to mass, we are frequently dealing with larger systems. More mass means a greater amount of heat is typically required to achieve the same change in temperature compared to a smaller mass of the same substance.
It is important to note:

• Proportionality: As mass increases, the total heat required to change the temperature by a degree also increases if specific heat and temperature change are constant.
• Uniformity: For accurate calculations, the mass treated must be uniform, meaning there are no variations in sample composition affecting temperature change response.
• Influence on Specific Heat: Specific heat is essentially independent of mass; it is a substance-specific value. However, when calculating total heat transfer, mass plays a critical role.

With these considerations, the formula \( q = mc\Delta T \) neatly captures the relationship between heat added, specific heat, mass, and temperature change, enabling us to solve for the specific heat capacity of a given substance.