Calculating the final temperature (\(T_f\)) of a substance is a fundamental part of thermodynamics, especially when dealing with heat transfer. Specific heat capacity is a physical property that quantifies the thermal energy required to change a substance’s temperature, measuring its resistance to temperature change. Problems requiring \(T_f\) fall into two main categories: those involving a known amount of energy transfer, and those where two substances at different temperatures are mixed. Both scenarios rely on the core principles of energy conservation.
Understanding the Specific Heat Formula
The relationship between heat transfer and temperature change is mathematically described by the specific heat formula: \(Q = mc\Delta T\). \(Q\) represents the heat energy absorbed (positive \(Q\)) or released (negative \(Q\)), measured in Joules (J). The mass of the substance, \(m\), is typically measured in grams (g) or kilograms (kg).
The specific heat capacity, \(c\), is an intrinsic property measured in units like \(J/g^\circ C\) or \(J/kg \cdot K\). \(\Delta T\) is the change in temperature, calculated by subtracting the initial temperature from the final temperature: \(\Delta T = T_{final} – T_{initial}\). Since the size of a degree Celsius is the same as a Kelvin, temperature changes can be expressed using either unit.
Isolating the Final Temperature Algebraically
To find the final temperature (\(T_f\)), the standard specific heat equation \(Q = mc\Delta T\) must be rearranged. Expanding \(\Delta T\) gives \(Q = mc(T_f – T_i)\).
To isolate \(T_f\), first divide both sides by \(mc\), resulting in \(\Delta T = Q / mc\). Since \(\Delta T = T_f – T_i\), the equation becomes \(T_f – T_i = Q / mc\). Adding the initial temperature (\(T_i\)) to both sides yields the simplified formula: \(T_f = T_i + (Q / mc)\). This expression is used when mass, specific heat, initial temperature, and transferred heat energy are known.
Calculating \(T_f\) When Energy Transfer Is Known
This scenario involves a single substance where a specific amount of heat energy (\(Q\)) is added or removed. The process begins by listing the known values for \(T_i\), \(m\), \(c\), and \(Q\). It is important to ensure that all units are consistent, such as using grams with \(J/g^\circ C\) and Joules for \(Q\).
The formula \(T_f = T_i + (Q / mc)\) is then used directly. If the substance is heated, \(Q\) is positive, and \((Q/mc)\) represents the temperature increase. If the substance is cooled, \(Q\) must be entered as a negative number, resulting in a negative temperature change and a final temperature lower than \(T_i\).
For example, if 100 grams of water (\(c \approx 4.184 J/g^\circ C\)) at \(20^\circ C\) absorbs 4,184 Joules of heat, the calculation is \(T_f = 20^\circ C + (4,184 J / (100 g \cdot 4.184 J/g^\circ C))\). The temperature change is \(10^\circ C\), resulting in \(T_f = 30^\circ C\). If the same heat were removed (\(Q = -4,184 J\)), the final temperature would be \(10^\circ C\). This demonstrates how the sign of \(Q\) dictates the direction of the temperature change.
Determining \(T_f\) When Two Substances Mix
This complex scenario, known as calorimetry, involves mixing two substances at different initial temperatures until they reach a single final equilibrium temperature (\(T_f\)). This relies on the conservation of energy, asserting that the heat lost by the warmer substance must equal the heat gained by the cooler substance, expressed as \(Q_1 + Q_2 = 0\).
To solve for \(T_f\), the specific heat formula is applied to both substances and set equal to zero: \((m_1 c_1 \Delta T_1) + (m_2 c_2 \Delta T_2) = 0\). The crucial step is substituting \(\Delta T\) for each substance using the common final temperature \(T_f\): \(m_1 c_1 (T_f – T_{i1}) + m_2 c_2 (T_f – T_{i2}) = 0\). Since \(T_f\) is the only unknown, the equation can be algebraically solved.
For instance, when mixing a hot metal (Substance 1) with cold water (Substance 2), the terms are expanded. All terms containing \(T_f\) are grouped on one side, and all known initial energy terms are grouped on the opposite side. \(T_f\) is then found by dividing the collected initial energy terms by the sum of the heat capacity products (\(m_1 c_1 + m_2 c_2\)).