Problem 105 - Physics

Question

In Section \(12.3\) it was mentioned that temperatures are often measured with electrical resistance thermometers made of platinum wire. Suppose that the resistance of a platinum resistance thermometer is \(125 \Omega\) when its temperature is \(20.0^{\circ} \mathrm{C}\). The wire is then immersed in boiling chlorine, and the resistance drops to \(99.6 \Omega\). The temperature coefficient of resistivity of platinum is \(\alpha=3.72 \times 10^{-3}\left(\mathrm{C}^{\circ}\right)^{-1}\). What is the temnerature of the boiling chlorine?

Step-by-Step Solution

Answer

The temperature of the boiling chlorine is approximately \(-34.6^{\circ} \mathrm{C}\).

Step 1: Understand the Problem

The problem involves determining the temperature of boiling chlorine using changes in electrical resistance with temperature and the temperature coefficient of resistivity for platinum.

Step 2: Use the Formula for Resistance and Temperature Relationship

The resistance of a platinum wire changes with temperature according to the formula: \[ R(T) = R_0 (1 + \alpha \Delta T) \] where \( R(T) \) is the resistance at temperature \( T \), \( R_0 \) is the initial resistance at a known temperature, \( \alpha \) is the temperature coefficient of resistivity, and \( \Delta T \) is the change in temperature.

Step 3: Substitute Known Values

From the problem, \( R_0 = 125 \Omega \) at \( T_0 = 20.0^{\circ} \mathrm{C} \), \( \alpha = 3.72 \times 10^{-3} \) \((\mathrm{C}^{\circ})^{-1}\), and \( R(T) = 99.6 \Omega \). Substitute these values into the formula: \[ 99.6 = 125 (1 + 3.72 \times 10^{-3} \Delta T) \]

Step 4: Solve for the Change in Temperature \( \Delta T \)

Rearrange the equation to solve for \( \Delta T \): \[ 99.6 = 125 + 125 \times 3.72 \times 10^{-3} \times \Delta T \] This simplifies to: \[ 99.6 - 125 = 125 \times 3.72 \times 10^{-3} \times \Delta T \]\[ -25.4 = 125 \times 3.72 \times 10^{-3} \times \Delta T \]\[ \Delta T = \frac{-25.4}{125 \times 3.72 \times 10^{-3}} \] After calculating, \( \Delta T \approx -54.57 \circ \mathrm{C} \).

Step 5: Determine the Boiling Temperature of Chlorine

The boiling temperature of chlorine is calculated as \( T = T_0 + \Delta T \).\[ T = 20.0^{\circ} \mathrm{C} - 54.57 \circ \mathrm{C} \] which gives \( T \approx -34.57 \circ \mathrm{C} \).

Key Concepts

Platinum Resistance Thermometer

A platinum resistance thermometer is a common device used for measuring temperature through electrical resistance. It primarily utilizes platinum, a well-known metal, due to its stability and linear resistance-temperature relationship. This linear relationship makes platinum ideal for precise temperature measurements over a wide range of temperatures. In essence, as the temperature of the platinum wire increases or decreases, its electrical resistance changes predictably. Here are some key points about this thermometer:

• Stable and Reliable: Platinum exhibits consistent and predictable changes in resistance with temperature changes.
• Wide Temperature Range: This type of thermometer can effectively measure a broad range of temperatures, from very low to very high.
• Accuracy: With its linearity and stability, it provides accurate and repeatable readings.

Understanding the role of platinum resistance thermometers is crucial when dealing with precise temperature measurements. They are widely used in science and industry due to their reliability and capacity to withstand extreme conditions.

Temperature Coefficient of Resistivity

The temperature coefficient of resistivity is a fundamental property of materials, including platinum, that describes how resistivity changes with temperature. It is denoted by the Greek letter alpha (\( \alpha \)). This coefficient indicates the fractional change in resistivity per degree Celsius change in temperature.

• Importance: The coefficient determines how significantly a material's resistance will increase or decrease with temperature changes.
• Platinum's Behavior: For platinum, this coefficient is particularly high, making changes in its resistance easily measurable for use in platinum resistance thermometers.
• Calculation: It is essential in equations that relate resistance and temperature, allowing calculation of unknown temperatures, as seen in this exercise.

A clear grasp of the temperature coefficient of resistivity lets us better understand how materials behave under varying temperatures, which is pivotal in designing and utilizing devices like resistance thermometers.

Resistance and Temperature Relationship

The relationship between electrical resistance and temperature is vital for understanding how devices like platinum resistance thermometers work. In the case of metals such as platinum, their resistance typically increases as temperature rises.The basic formula connecting resistance (\( R \)) to temperature (\( T \)) is given by:\[R(T) = R_0 (1 + \alpha \Delta T)\]- \( R(T) \) is the resistance at temperature \( T \)- \( R_0 \) is the initial resistance at a known temperature- \( \alpha \) is the temperature coefficient of resistivity- \( \Delta T \) is the change in temperature.By using this formula, we can determine how much the resistance changes for a given temperature change, which is crucial when calculating unknown temperatures, such as determining the boiling point of a liquid by immersing a resistance thermometer in it. The linearity of this relationship makes it feasible to derive temperatures over various ranges accurately. Understanding this concept allows engineers and scientists to design systems capable of consistently accurate thermal measurements and controls.