What Is The Final Temperature When A 5.0 G Sample Of Brass Initially At 25°C Absorbs 75 J Of Heat? The
Understanding the thermal response of materials to heat transfer is a fundamental concept in thermodynamics and materials science. When a specific amount of heat is supplied to a substance, its temperature changes depending on properties such as specific heat capacity, mass, and initial temperature. In this article, we will explore in detail how to determine the final temperature of a small brass sample after it absorbs a given quantity of heat, elucidate the underlying principles, and provide a comprehensive step-by-step calculation.
Fundamentals of Heat Transfer and Specific Heat Capacity
What Is Heat and How Is It Transferred?
Heat is a form of energy transfer between systems or objects due to temperature differences. It can be transferred through conduction, convection, or radiation. When heat is added to a substance, it increases the internal energy, leading to a rise in temperature assuming no phase change occurs.Specific Heat Capacity: Definition and Significance
The specific heat capacity (c) of a material is the amount of heat required to raise the temperature of 1 gram of the substance by 1°C (or 1 K). It is expressed in units of J/(g·°C) or J/(g·K). Different materials have varying specific heat capacities; for example, metals typically have lower specific heat capacities than water.Heat Equation for Temperature Change
The fundamental relation connecting heat transfer and temperature change is:\[ Q = mc\Delta T \]
where:
- \( Q \) = heat added or removed (J)
- \( m \) = mass of the substance (g)
- \( c \) = specific heat capacity (J/(g·°C))
- \( \Delta T \) = change in temperature (°C)
This equation assumes no phase change and constant specific heat capacity over the temperature range.
Properties of Brass Relevant to this Problem
What Is Brass?
Brass is an alloy primarily composed of copper and zinc, and it exhibits specific thermodynamic properties that are well documented. The specific heat capacity of brass typically varies around 0.380 J/(g·°C). For precise calculations, consulting a reliable data source or material property chart is recommended.Typical Specific Heat Capacity of Brass
- Approximate value: \( c_{brass} \approx 0.380 \) J/(g·°C)
Mass and Initial Temperature
- Mass, \( m \): 5.0 g
- Initial temperature, \( T_{initial} \): 25°C
Calculating the Final Temperature
Step 1: Write Down Known Quantities
- Heat absorbed, \( Q \): 75 J
- Mass, \( m \): 5.0 g
- Specific heat capacity, \( c \): 0.380 J/(g·°C)
- Initial temperature, \( T_{initial} \): 25°C
Step 2: Rearrange the Heat Equation to Find \( \Delta T \)
\[ \Delta T = \frac{Q}{mc} \]This formula calculates the temperature change resulting from the absorbed heat.
Step 3: Compute \( \Delta T \)
\[ \Delta T = \frac{75\, \text{J}}{5.0\, \text{g} \times 0.380\, \text{J/(g·°C)}} = \frac{75}{1.9} \approx 39.47^\circ C \]The temperature increase is approximately 39.47°C.
Step 4: Determine the Final Temperature
\[ T{final} = T{initial} + \Delta T = 25^\circ C + 39.47^\circ C \approx 64.47^\circ C \]Thus, after absorbing 75 J of heat, the brass sample’s temperature will rise to approximately 64.5°C.
Discussion and Considerations
Assumption of Constant Specific Heat Capacity
The calculation assumes that the specific heat capacity remains constant over the temperature range. In reality, \( c \) can vary slightly with temperature, but for small temperature changes like this, the approximation is typically valid.Neglecting Heat Losses
The calculation presumes no heat loss to surroundings during the process, which is idealized. In practical scenarios, some heat may dissipate, leading to a slightly lower final temperature.Phase Changes and Material Integrity
Since the temperature increase is well below brass’s melting point (around 900°C), phase changes are not considered. The material remains in the solid state.Additional Factors Influencing Final Temperature
- Environmental Conditions: Insulation or heat exchange with surroundings affects actual temperature change.
- Heat Capacity Variability: For precise calculations, temperature-dependent specific heat capacities can be used instead of a constant value.
- Measurement Precision: Error margins in mass, heat input, and specific heat data can influence results.
Summary
To summarize, the final temperature of a 5.0 g brass sample initially at 25°C, after absorbing 75 J of heat, is approximately 64.5°C. This calculation hinges on fundamental principles of thermodynamics, assuming constant specific heat capacity and no heat losses. Such calculations are essential in designing thermal systems, understanding material behavior under heating, and in various engineering applications.
Conclusion
Understanding how heat affects materials is crucial across scientific and engineering disciplines. By applying the heat equation and knowledge of material properties, we can predict temperature changes with reasonable accuracy. In this specific case involving brass, the process demonstrates how relatively small quantities of heat can significantly raise the temperature of a small metal sample, highlighting the importance of specific heat capacity and mass in thermal calculations. This knowledge is not only academically interesting but also practically valuable in manufacturing, metallurgy, and thermal management systems.
