Question

Why is it incorrect to say that a sample of helium at \(50^{\circ} \mathrm{C}\) is twice as hot as a sample of helium at \(25^{\circ} \mathrm{C} ?\)

Short answer

It is incorrect to say that a sample of helium at \(50^{\circ}\mathrm{C}\) is twice as hot as a sample of helium at \(25^{\circ}\mathrm{C}\) because the Celsius scale is a relative scale. To make accurate comparisons using ratios, we need to convert the temperatures to an absolute scale like Kelvin. After conversion, we get \(T_{50^{\circ}\mathrm{C}} = 323.15\,\mathrm{K}\) and \(T_{25^{\circ}\mathrm{C}} = 298.15\,\mathrm{K}\), yielding a temperature ratio of \(\approx 1.08\), which is not equal to 2.

Step-by-step solution

Conversion of temperatures to Kelvin scale

To convert the given temperatures from Celsius to Kelvin, we need to add 273.15 to each of them: \[ T_{50^{\circ}\mathrm{C}} = 50 + 273.15 = 323.15\,\mathrm{K}\\ T_{25^{\circ}\mathrm{C}} = 25 + 273.15 = 298.15\,\mathrm{K} \]

Comparing temperatures in Kelvin scale

Now that we have the temperatures in Kelvin, we can compare them to see if the statement "a sample of helium at \(50^{\circ} \mathrm{C}\) is twice as hot as a sample of helium at \(25^{\circ}\mathrm{C}\)" is correct. Let's calculate the ratio of temperatures: \[ \frac{T_{50^{\circ}\mathrm{C}}}{T_{25^{\circ}\mathrm{C}}} = \frac{323.15\,\mathrm{K}}{298.15\,\mathrm{K}} \approx 1.08 \] Since the ratio of the temperatures is not equal to 2, the statement mentioned is incorrect.

Key concepts

Kelvin Scale

The Kelvin scale is an absolute temperature scale used within the scientific community for a precise measurement of thermodynamic temperature. Unlike Celsius or Fahrenheit, the Kelvin scale starts at absolute zero—the point at which all thermal motion ceases in the classical description of thermodynamics. This is 0 Kelvin (0 K).

Kelvin is preferred for scientific calculations because it avoids the confusion that arises from the negative temperatures found on other scales. It is closely related to the Celsius scale: each increment on the Kelvin scale is equivalent to a degree on the Celsius scale, but it shifts the starting point. Temperature in Kelvin can be obtained from Celsius using the formula:

• Kelvin (K) = Celsius (°C) + 273.15

This conversion ensures that calculations involving ratios of temperatures are straightforward and consistent.

Celsius to Kelvin Conversion

When converting temperatures from the Celsius scale to the Kelvin scale, it's important to use a simple formula to ensure accuracy. This process is crucial for calculations that require comparisons between temperatures.

The formula used for this conversion is:

• Kelvin (K) = Celsius (°C) + 273.15

For example, to convert 25°C into Kelvin, we add 273.15, resulting in 298.15 K. Similarly, converting 50°C yields 323.15 K. This method allows us to work with absolute temperatures, which are necessary for certain scientific analyses and comparisons. Using the Kelvin scale eliminates the complications of dealing with zero or negative numbers, which cannot be meaningfully manipulated in multiplication or division.

Temperature Comparison

When comparing temperatures, particularly when discussing whether one temperature is proportionally higher than another, it is best to use the Kelvin scale due to its absolute nature.

In scenarios like the exercise question—determining if 50°C is twice as hot as 25°C—the initial thought may be misleading. After converting both temperatures to Kelvin (323.15 K for 50°C and 298.15 K for 25°C), it becomes clear through basic division that the ratio is approximately 1.08, not 2.

The error arises from not considering the baseline of the temperature scale being used. In Celsius, degrees are measured relative to the freezing point of water, whereas Kelvin is measured from absolute zero. This distinction impacts how we interpret temperature quantities and how they relate multiplicatively.