Question

A thermometer gives a reading of \(96.1^{\circ} \mathrm{F} \pm 0.2^{\circ} \mathrm{F}\). What is the temperature in \({ }^{\circ} \mathrm{C}\) ? What is the uncertainty?

Short answer

The temperature in Celsius is approximately \(35.6^{\circ} \mathrm{C}\) with an uncertainty of \(\pm 0.1^{\circ} \mathrm{C}\).

Step-by-step solution

Convert the temperature in Fahrenheit to Celsius

To convert the temperature from Fahrenheit to Celsius, use the formula: \(T_{C}=\frac{5}{9}(T_{F}-32)\). Plug in the given temperature in Fahrenheit, \(96.1^{\circ} \mathrm{F}\), into the formula: \[T_{C}=\frac{5}{9}(96.1-32)\] Calculate the temperature: \(T_{C}\approx 35.6^{\circ} \mathrm{C}\) The temperature in Celsius is approximately \(35.6^{\circ} \mathrm{C}\).

Calculate the uncertainty

To find the uncertainty in the Celsius temperature, we can use the derivative of the conversion formula with respect to the Fahrenheit temperature and multiply by the uncertainty in Fahrenheit (error propagation): \[\Delta T_{C} = \left|\frac{d}{dT_{F}}\left(\frac{5}{9}(T_{F}-32)\right)\right| \cdot \Delta T_{F}\] First, differentiate the conversion formula with respect to Fahrenheit temperature: \[\frac{d}{dT_{F}}\left(\frac{5}{9}(T_{F}-32)\right) = \frac{5}{9}\] Next, multiply the derivative by the given uncertainty in Fahrenheit (\(\pm 0.2^{\circ} \mathrm{F}\)): \[\Delta T_{C} = \left|\frac{5}{9}\right| \cdot 0.2 \approx 0.1\] The uncertainty in the Celsius temperature is approximately \(\pm 0.1^{\circ} \mathrm{C}\).

Write the final answer

Combine the calculated temperature in Celsius and its uncertainty: \(T_{C} = 35.6^{\circ} \mathrm{C} \pm 0.1^{\circ} \mathrm{C}\) The temperature in Celsius is approximately \(35.6^{\circ} \mathrm{C}\) with an uncertainty of \(\pm 0.1^{\circ} \mathrm{C}\).

Key concepts

Fahrenheit to Celsius Conversion

Understanding how to convert temperature readings from Fahrenheit to Celsius is a fundamental skill in science and daily life, especially when you're dealing with international temperature scales. The formula for this conversion is relatively simple:

• To convert temperature from Fahrenheit (\(T_F\text{ in }^{\text{\textdegree}}F\text{)\text{ to Celsius (\text{Celsius}})\text{, the formula is:}\[T_C = \frac{5}{9}(\ T_F - 32\)\]\
• The number 32 in this formula reflects the Fahrenheit temperature of the freezing point of water, which corresponds to 0\text{ºC}.
• The fraction \(\frac{5}{9}\)\text{ represents the ratio of the temperature scales between Celsius and Fahrenheit.}

Let's take the example where a thermometer gives a reading of \(96.1^{\text{\textdegree}} F \pm 0.2^{\text{\textdegree}}F\). By applying the formula, we subtract 32 from the Fahrenheit temperature and then multiply by \(\frac{5}{9}\) to convert it into Celsius. The result gives us the correct temperature in Celsius.

Error Propagation

When converting measurements, it's crucial to understand that uncertainties in data, often referred to as measurement errors, can affect the final result. This is where the concept of error propagation comes into play. Error propagation is the process of calculating the uncertainty of an outcome resulting from multiple measurements, each with their own uncertainties.

Calculating Error Propagation in Temperature Conversion

To find the propagated error when converting Fahrenheit to Celsius, we use the derivative of the conversion formula with respect to Fahrenheit since the derivative gives the rate at which the conversion factor changes. In our example, the derivative is constant, \(\frac{5}{9}\). We then multiply this rate of change by the uncertainty in the Fahrenheit measurement to estimate the uncertainty in Celsius:

\[\Delta T_C = \left|\frac{d}{dT_F} \left(\frac{5}{9}(T_F - 32) \right) \right| \cdot \Delta T_F\]\
For our given temperature reading with an uncertainty of \(\pm 0.2^{\text{\textdegree}}F\), the calculation yields an uncertainty in Celsius that needs to be accounted for in the final result. Error propagation ensures that the final reported measurement accurately represents the range in which the true value likely exists.

Uncertainty Calculation

Uncertainty calculation is a vital aspect of any scientific measurement—it quantifies the doubt about the measurement result. Whenever a measurement is made in science – whether it's temperature, length, or volume – there's always a degree of uncertainty associated with it due to potential measurement errors, instrument quality or calibration, environmental conditions, and other factors.

Estimating Uncertainty in Temperature Measurements

Using our thermometer example, the temperature is reported as \(96.1^{\text{\textdegree}} F \pm 0.2^{\text{\textdegree}} F\). The \(\pm\) sign represents the bounds of uncertainty around the measurement. After converting to Celsius, we estimated the new uncertainty as \(\pm 0.1^{\text{\textdegree}} C\). This uncertainty calculation provides a range (\(35.5^{\text{\textdegree}} C \text{ to } 35.7^{\text{\textdegree}} C\)) that likely contains the true temperature. It's important to report this uncertainty along with the measured value so that anyone relying on this data understands its accuracy and limits.
Conveying the bounds of measurement accuracy helps to build trust in scientific data and supports informed decision-making. Uncertainty calculation isn't an afterthought; it's a key part of the measurement that reflects honest and precise scientific communication.