In the table below, a number in the top row represents the pressure of a gas in the bulb of a constant-volume gas thermometer (corrected for dead space, thermal expansion of bulb, etc.) when the bulb is immersed in a water triple- point cell. The bottom row represents the corresponding readings of pressure when the bulb is surrounded by a material at a constant unknown temperature. Calculate the idealgas temperature \(T\) of this material. (Use five significant figures.) \(P_{T P}, \mathrm{kPa} \quad 133.32\) \(\begin{array}{lll}99.992 & 66.661 & 33.331\end{array}\) \(P, \mathrm{kPa}\) \(204.69 \quad 153.54\) \(102.37\) \(51.190\)

A constant-volume gas thermometer is a device used to measure temperature. It relies on the property of gases to expand and contract with changes in temperature. This thermometer consists of a gas-filled bulb connected to a mercury manometer. The volume of gas in the bulb is kept constant by adjusting the mercury level.

When the temperature changes, the pressure of the gas also changes. By measuring the pressure of the gas at known temperatures, such as the triple-point of water, we can use these reference readings to determine the temperature of an unknown environment. The pressure readings at different temperatures are used to calculate the unknown temperatures using the ideal gas law.

This method is highly accurate and widely used in calibration laboratories.

The triple-point temperature of a substance is the unique temperature at which all three phases (solid, liquid, and gas) coexist in thermodynamic equilibrium. For water, this temperature is exactly 273.16 K (0.01°C).

This point is crucial for thermometer calibration because it provides a precise reference standard. The pressure of a gas at the triple-point temperature is used to calculate temperatures in a constant-volume gas thermometer. Quality standard methods like the triple-point of water ensure high accuracy in temperature measurement.

When dealing with high-precision thermometry, understanding and utilizing the triple-point temperature of water is essential. It serves as a fixed and universal benchmark for thermometer calibration.

The ideal gas law is a fundamental equation in thermodynamics that describes the behavior of an ideal gas. It is expressed as: This law represents the relationship between pressure (P), volume (V), temperature (T), and the number of moles of gas (n), with R being the gas constant (8.314 J/mol·K).

In a constant-volume gas thermometer, the volume (V) is kept constant. Hence, the ideal gas law simplifies to the form: , where pressure is directly proportional to temperature.

This proportionality is leveraged to calculate the unknown temperature when the gas pressure is measured. Understanding the ideal gas law is crucial for accurately determining temperatures using constant-volume gas thermometers.

Understanding the temperature-pressure relationship is fundamental for working with gases. According to the ideal gas law, at constant volume, the pressure of a gas is directly proportional to its absolute temperature. This means as the temperature increases, the pressure increases, and vice versa.

Mathematically, this relationship is expressed as: , where T is the temperature in Kelvin and P is the pressure.

When using a constant-volume gas thermometer, this relationship allows us to measure the pressure at the unknown temperature and compare it to the pressure at a known reference temperature, like the triple-point of water. From this, the unknown temperature can be calculated by combining the known value of the triple-point temperature and the measured pressures.

This temperature-pressure relationship simplifies the process of temperature measurement using ideal gas properties and makes it an accurate method for various scientific and industrial applications.