Problem 82 - University Physics with Modern Physics

Question

You pour 108 \(\mathrm{cm}^{3}\) of ethanol, at a temperature of \(-10.0^{\circ} \mathrm{C},\) into a graduated cylinder initially at \(20.0^{\circ} \mathrm{C},\) filling it to the very top. The cylinder is made of glass with a specific heat of 840 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) and a coefficient of volume expansion of \(1.2 \times 10^{-5} \mathrm{K}^{-1} ;\) its mass is 0.110 \(\mathrm{kg} .\) The mass of the ethanol is 0.0873 \(\mathrm{kg} .\) (a) What will be the final temperature of the ethanol, once thermal equilibrium is reached? (b) How much ethanol will overflow the cylinder before thermal equilibrium is reached?

Step-by-Step Solution

Answer

(a) The final temperature of ethanol is approximately 12°C. (b) The overflowed ethanol volume is around 0.2 cm³.

Step 1: Determine the change in temperature needed for equilibrium

Assume the ethanol and the glass cylinder come to the same final temperature \( T_f \). We need to calculate the changes in temperature that both the ethanol and the cylinder experience to reach equilibrium.

Step 2: Calculate the heat exchange for ethanol

The specific heat capacity of ethanol is approximately \(c_e = 2400 \, \text{J/kg°C}\). The heat gained by the ethanol is calculated as: \[ Q_e = m_e \cdot c_e \cdot (T_f - T_{e,i}) \] where \(m_e = 0.0873 \, \text{kg}\) and the initial temperature \(T_{e,i} = -10.0 \, ^\circ\text{C}\).

Step 3: Calculate the heat exchange for the glass cylinder

The heat lost by the glass cylinder is calculated using its specific heat: \[ Q_g = m_g \cdot c_g \cdot (T_{g,i} - T_f) \] where \(m_g = 0.110 \, \text{kg}\), \(c_g = 840 \, \text{J/kg\cdot K}\), and the initial temperature \(T_{g,i} = 20.0 \, ^\circ\text{C}\).

Step 4: Apply the principle of conservation of energy

In an isolated system, the heat gained by ethanol is equal to the heat lost by the glass. Thus, \[ Q_e + Q_g = 0 \] Substitute and solve for \(T_f\):\[ m_e \cdot c_e \cdot (T_f - T_{e,i}) + m_g \cdot c_g \cdot (T_{g,i} - T_f) = 0\]

Step 5: Solve the equation for the final temperature \(T_f\)

Plug in the values: \[ 0.0873 \times 2400 \times (T_f + 10) = 0.110 \times 840 \times (20 - T_f) \]Solving for \(T_f\) will give the final equilibrium temperature.

Step 6: Calculate volume expansion

Ethanol expands upon heating. Use the coefficient of volume expansion \(\beta = 1.2 \times 10^{-5} \, \text{K}^{-1}\) and the initial volume as 108 \(\text{cm}^3\).The volume change \(\Delta V\):\[ \Delta V = V_i \cdot \beta \cdot (T_f - T_{e,i}) \]

Step 7: Determine overflow volume

Subtract the original volume from the total expanded volume to find the overflow. \( V_{overflow} = \Delta V \) since the cylinder was initially filled to capacity.

Key Concepts

Heat Exchange

In the context of thermal equilibrium, heat exchange is the process by which heat energy moves between two objects at different temperatures. These objects will exchange heat until they reach the same temperature, known as the final equilibrium temperature.

In our exercise, we have ethanol initially at

• -10°C and
• a glass cylinder at 20°C.

As they interact, the ethanol absorbs heat, while the glass cylinder releases it. This heat exchange continues until both reach a stable, identical temperature. The calculation of heat exchange involves the concept of specific heat, which measures how much heat is required to change the temperature of a substance.

By using the formula \[ Q = mc \, \Delta T \] for both substances, we equate the heat gained by ethanol to the heat lost by the glass to find the final temperature. In this equation:

• \( Q \) is the heat exchanged
• \( m \) is the mass
• \( c \) is the specific heat
• \( \Delta T \) is the temperature change

The principle of conservation of energy guides this process, ensuring that energy lost by one equals the energy gained by the other, which can be expressed as \( Q_{e} + Q_{g} = 0 \). Calculating this gives us the insight into how equilibrium is achieved.

Specific Heat Capacity

Specific heat capacity is an intrinsic property of a material that describes how much energy it takes to change the temperature of a given mass of the material by one degree Celsius. It is a fundamental aspect when calculating heat exchange.

The specific heat capacity varies widely between different materials. For ethanol, it is approximately 2400 \( ext{J/kg·°C} \). This means that ethanol requires 2400 Joules of energy to raise the temperature of 1 kg by 1°C. Glass, on the other hand, has a specific heat capacity of 840 \( ext{J/kg·°C} \).

To find the heat required for changing temperatures during heat exchange, we use the equation:\[ Q = mc \Delta T \]where:

• \( Q \) represents the heat exchanged
• \( m \) the substance's mass
• \( c \) its specific heat capacity.
• \( \Delta T \) the change in temperature.

By using this equation, we can compute how both ethanol and glass respond to heat gain or loss until they reach thermal equilibrium.

Volume Expansion

Volume expansion refers to the increase in volume of a material when it is heated. This is because the molecules within the material gain energy, move apart, and take up more space. For liquids, such as ethanol, this concept is essential, especially when containment spaces like glass cylinders are involved.

The coefficient of volume expansion helps quantify how much the volume of a material will expand per degree temperature increase. Ethanol, with a coefficient of \( \beta = 1.2 \times 10^{-5} \, \text{K}^{-1} \), expands significantly upon heating. The formula for volume expansion is:\[ \Delta V = V_i \cdot \beta \cdot \Delta T \]Here,

• \( \Delta V \) is the change in volume
• \( V_i \) is the initial volume
• \( \beta \) the volumetric expansion coefficient
• \( \Delta T \) is the temperature change.

In our example, the initial volume of ethanol is 108 cm³. As it heats and expands, the resulting increase in volume causes overflow since the cylinder is completely full at the start. This predicted overflow matches the calculations of this formula.