Problem 45 - General Chemistry: Principles and Modern Applications

Question

You decide to establish a new temperature scale on which the melting point of mercury \(\left(-38.9^{\circ} \mathrm{C}\right)\) is \(0^{\circ} \mathrm{M},\) and the boiling point of mercury \(\left(356.9^{\circ} \mathrm{C}\right)\) is \(100^{\circ} \mathrm{M} .\) What would be (a) the boiling point of water in \(^{\circ} \mathrm{M} ;\) and \((\mathrm{b})\) the temperature of absolute zero in \(^{\circ}\text{M}\)?

Step-by-Step Solution

Answer

To answer the results are as follows: (a) The boiling point of water in M degrees is just the value you obtained from substituting T_C = 100 into the equation. (b) The absolute zero temperature in M degrees is the result you got from substituting T_C = -273.15 into the equation applied.

Step 1: Define the linear equation for the conversion

First, form a linear equation using the given mercury points. Since we know that at -38.9°C, the temperature is 0M, and at 356.9°C, it's 100M, the equation will be in the form: \(T_M = a T_C + b\). But for T_C = -38.9°C, T_M = 0M and for T_C = 356.9°C, T_M = 100M.

Step 2: Calculation of the equation coefficients

To find the coefficients a and b, we'll substitute the given values into the equation. Starting with the known point (-38.9,0), we substitute these as T_C and T_M in the equation, obtaining: \(0 = a*(-38.9) + b\). And then, using the point (356.9,100): \(100=a*356.9 +b\). Solving these simultaneous equations gives us the values of a and b for the equation.

Step 3: Calculating temperatures in Mercury scale

With the linear equation established, we can find the Mercury equivalent for any Celsius temperature. For example, to solve for part (a), where the boiling point of water is 100°C, substitute T_C = 100 into the equation to find T_M. For part (b), consider that absolute zero is -273.15°C in the Celsius scale, so substitute T_C = -273.15 to find T_M.

Key Concepts

Linear Equation Derivation

When creating a new temperature scale, it's crucial to define how the temperatures will convert from one scale to the other. This conversion is typically represented by a linear equation of the form:

• \( T_M = a \cdot T_C + b \)

where \( T_M \) is the temperature in the new scale (Mercury Scale in this case), \( T_C \) is the temperature in Celsius, \( a \) is the slope of the line, and \( b \) is the y-intercept. The slope \( a \) represents how much \( T_M \) changes for a unit change in \( T_C \), while \( b \) indicates where the line crosses the \( T_M \) axis.
To derive this equation, we use two known points: the melting point of mercury \((-38.9^{\circ} \text{C}, 0^{\circ} \text{M})\) and its boiling point \( (356.9^{\circ} \text{C}, 100^{\circ} \text{M})\). These points give us a system of equations:

• \( 0 = a \cdot (-38.9) + b \)
• \( 100 = a \cdot 356.9 + b \)

Solving these equations simultaneously, we can find the specific values for \( a \) and \( b \). First, isolate \( b \) in the first equation: \( b = 38.9a \). Substitute \( b \) in the second equation and solve for \( a \). Once \( a \) is known, plug back to find \( b \). This will give us the linear conversion equation needed for any temperature conversion from Celsius to the Mercury scale.

Melting and Boiling Points

Temperature scales, like the newly created Mercury Scale, need defined reference points for consistency. For most scales, the melting and boiling points of a substance are chosen because they are fixed and repeatable. These points determine the scale's intervals.
For the Mercury Scale, the melting and boiling points of mercury were chosen, with \(-38.9^{\circ}\text{C}\) set to \(0^{\circ}\text{M}\) and \(356.9^{\circ}\text{C}\) set to \(100^{\circ}\text{M}\). By establishing these points:

• \(-38.9^{\circ}\) as the baseline or starting point (\(0^{\circ}\text{M}\)).
• \(356.9^{\circ}\) as the endpoint (\(100^{\circ}\text{M}\)).

This choice provides a direct method to map Celsius to Mercury. Other scales, like Celsius and Fahrenheit, use water for these reference points. In Celsius, the scale sets water's melting point at \(0^{\circ}\text{C}\) and boiling at \(100^{\circ}\text{C}\). Each scale provides similar thermal intervals, but with different starting and ending points, allowing for varied applications depending on the desired precision and use-case.

Absolute Zero Temperature

Absolute zero is a crucial reference in understanding temperature because it represents the theoretical lowest temperature possible. At this point, all molecular motion ceases, and it is equivalent to \(-273.15^{\circ}\text{C}\) in the Celsius scale.
Understanding this concept helps in expanding the Mercury Scale to measure a broader range of temperatures. Since absolute zero is a fundamental benchmark across all temperature scales, converting it into our Mercury Scale involves substituting \( T_C = -273.15 \) in our derived linear equation. By doing this, we find the Mercury Scale equivalent of the absolute zero point.
Absolute zero helps in studying various physical properties and behaviors of substances at extremely low temperatures. Although it is not achievable in practice, establishing it in a newly defined scale provides a complete understanding of temperature measurements from the coldest possible to hotter temperatures like the boiling point of mercury.