Question

A 237 -g piece of molybdenum, initially at \(100.0^{\circ} \mathrm{C},\) is dropped into \(244 \mathrm{g}\) of water at \(10.0^{\circ} \mathrm{C} .\) When the system comes to thermal equilibrium, the temperature is \(15.3^{\circ} \mathrm{C}\) What is the specific heat capacity of molybdenum?

Short answer

The specific heat capacity of molybdenum is approximately 0.270 J/g°C.

Step-by-step solution

Understand Heat Transfer Concept

When two substances at different temperatures are in thermal contact, heat will flow from the hotter to the cooler until they reach the same temperature, meaning they are in thermal equilibrium. The heat lost by the hot substance equals the heat gained by the cold substance.

Identify Known Quantities

We have a 237-g piece of molybdenum and its initial and final temperatures are known: initial is \(100.0^{\circ} \mathrm{C}\), final is \(15.3^{\circ} \mathrm{C}\). We also have 244 g of water with initial temperature \(10.0^{\circ} \mathrm{C}\) and final temperature \(15.3^{\circ} \mathrm{C}\). The specific heat capacity of water \(c_{\text{water}}\) is \(4.18\, \text{J/g}^{\circ}\text{C}\). Specific heat capacity of molybdenum \(c_{\text{Mo}}\) is what we're trying to find.

State Heat Transfer Equations

The heat lost by molybdenum and gained by water can be expressed as: \[ q_{\text{Mo}} = m_{\text{Mo}} \cdot c_{\text{Mo}} \cdot (T_{\text{final}} - T_{\text{initial, Mo}}) \] \[ q_{\text{water}} = m_{\text{water}} \cdot c_{\text{water}} \cdot (T_{\text{final}} - T_{\text{initial, water}}) \] Equating the heat lost and gained gives: \[ m_{\text{Mo}} \cdot c_{\text{Mo}} \cdot (T_{\text{final}} - T_{\text{initial, Mo}}) = m_{\text{water}} \cdot c_{\text{water}} \cdot (T_{\text{final}} - T_{\text{initial, water}}) \]

Substitute Known Values

Substitute the known values into the equation: \[ 237 \cdot c_{\text{Mo}} \cdot (15.3 - 100.0) = 244 \cdot 4.18 \cdot (15.3 - 10.0) \] Simplify both sides: \[ 237 \cdot c_{\text{Mo}} \cdot (-84.7) = 244 \cdot 4.18 \cdot 5.3 \]

Solve for Specific Heat Capacity of Molybdenum

First, calculate the right side: \[ 244 \cdot 4.18 \cdot 5.3 = 5415.932 \] Then solve for \(c_{\text{Mo}} \): \[ c_{\text{Mo}} = \frac{5415.932}{237 \cdot (-84.7)} \] \[ c_{\text{Mo}} = \frac{5415.932}{-20079.9} \] \[ c_{\text{Mo}} \approx 0.270 \text{J/g}^{\circ}\text{C} \] (taking absolute value as \(c_{\text{Mo}}\) must be positive).

Key concepts

Molybdenum

Molybdenum is a fascinating chemical element with the symbol Mo and atomic number 42. It is a transition metal that is highly valuable in various industrial applications.
Molybdenum is known for its robustness, resistant to heat and corrosion, making it very useful in high-temperature settings such as in the construction of spacecraft and industrial manufacturing.

• Origin: Molybdenum does not occur as a free element naturally, but is found in different oxidation states in minerals.
• Applications: It is used in steel alloys to enhance strength and resilience, thus being a critical component in machinery and equipment that operate under intense heat.
• Thermal Properties: Molybdenum has a relatively high melting point of 2623 °C, contributing to its heat-resistant properties and making it ideal for applications that require enduring high temperatures.

Understanding these properties can help explain why molybdenum is often the subject of physics problems regarding heat transfer, especially when calculating specific heat capacity under varying temperature conditions.

Heat Transfer

Heat transfer involves the movement of thermal energy from one object or material to another. This process is crucial whenever objects or systems at different temperatures interact. The basic principle is that heat will always flow from a warmer object to a cooler one until they reach an equal temperature.
There are three primary modes of heat transfer:

• Conduction: The transfer of heat through a material without any motion of the material itself. It occurs within a body or from one body to another in direct contact.
• Convection: The transfer of heat by the movement of fluids—liquids or gases. Warmer and cooler fluid masses move, enhancing the heat transfer process.
• Radiation: The transfer of energy by electromagnetic waves and does not require a medium, meaning it can happen through a vacuum.

In the case of molybdenum being placed in water in our exercise, heat transfer occurs largely by conduction as the solid metal and liquid are in direct contact. Additionally, the surroundings may allow for some heat loss by convection and radiation, though this is often minimal for insulated systems in short-term experiments.

Thermal Equilibrium

Thermal equilibrium is the state achieved when two or more objects in thermal contact reach the same temperature and no more heat energy is exchanged between them. This principle is a cornerstone of thermodynamics and plays an essential role in solving problems related to heat exchange.
When reaching thermal equilibrium:

• All components within the system share the same temperature.
• Heat transfer ceases because there is no longer a temperature difference to drive the flow of energy.
• The quantity of heat lost by the hot body is equal to the heat gained by the cooler body.

In our example, molybdenum and water start at different temperatures. As they interact, molybdenum loses heat while water gains heat, until reaching an equilibrium temperature of 15.3°C, the point where no additional heat transfer is observed. Understanding this concept helps in accurately calculating the specific heat capacity of materials by measuring temperature changes and equalizing energy exchanges.