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Chapter 21: Problem 6

At what temperature is the Fahrenheit scale reading equal to (a) twice that of the Celsius and \((b)\) half that of the Celsius?

Short Answer

Expert verified
The Fahrenheit scale reading is twice the Celsius reading at 160°C. There is no positive Celsius temperature where the Fahrenheit scale reading is half the Celsius reading.

Step by step solution

01

Find Equation For (a)

First, let's find the temperature at which the Fahrenheit scale reading equals twice the Celsius reading. Based on the given, if the Fahrenheit reading is twice the Celsius reading, var t stands for the required temperature in Celsius. So write down the relationship \( F = 2C \). Set equal to the Fahrenheit to Celsius conversion formula: \( 2C = 9/5C + 32 \).
02

Solve Equation For (a)

Solving the equation \( 2C = 9/5C + 32 \) for C. To do this, simplify by multiplying all terms by 5 to get rid of the fraction: \( 10C = 9C + 32 \times 5 \). Then isolate the C term by subtracting 9C from both sides: \( C = 160 \). So, at 160°C, the Fahrenheit reading is twice the Celsius reading.
03

Find Equation For (b)

Next, let's find the temperature at which the Fahrenheit scale reading equals half the Celsius reading. Using the same approach as (a), the relationship can be written as \( F = C/2 \). Also equal to the Fahrenheit to Celsius conversion: \( C/2 = 9/5C + 32 \).
04

Solve Equation For (b)

Solve the equation \( C/2 = 9/5C + 32 \). Again, multiply all terms by 5 to simplify: \( 5/2C = 9C + 32 \times 5 \). Subtracting 9C from both sides to isolate the C term results in a negative Celsius temperature. Thus, there are no positive Celsius temperature values for which the Fahrenheit reading equals half the Celsius reading. So, there is no solution for this part.

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Most popular questions from this chapter

An air bubble of \(19.4 \mathrm{~cm}^{3}\) volume is at the bottom of a lake \(41.5 \mathrm{~m}\) deep where the temperature is \(3.80^{\circ} \mathrm{C}\). The bubble rises to the surface, which is at a temperature of \(22.6^{\circ} \mathrm{C}\). Take the temperature of the bubble to be the same as that of the surrounding water and find its volume just before it reaches the surface.

Oxygen gas having a volume of \(1130 \mathrm{~cm}^{3}\) at \(42.0^{\circ} \mathrm{C}\) and a pressure of \(101 \mathrm{kPa}\) expands until its volume is \(1530 \mathrm{~cm}^{3}\) and its pressure is \(106 \mathrm{kPa}\). Find \((a)\) the number of moles of oxygen in the system and ( \(b\) ) its final temperature.

At \(100^{\circ} \mathrm{C}\) a glass flask is completely filled by \(891 \mathrm{~g}\) of mercury. What mass of mercury is needed to fill the flask at \(-35^{\circ} \mathrm{C} ?\) (The coefficient of linear expansion of glass is \(9.0 \times 10^{-6} / \mathrm{C}^{\circ} ;\) the coefficient of volume expansion of mercury is \(1.8 \times 10^{-4} / \mathrm{C}^{\circ}\).)

Density is mass divided by volume. If the volume \(V\) is temperature dependent, so is the density \(\rho\). Show that the change in density \(\Delta \rho\) with change in temperature \(\Delta T\) is given by $$\Delta \rho=-\beta \rho \Delta T$$ where \(\beta\) is the coefficient of volume expansion. Explain the minus sign.

A resistance thermometer is a thermometer in which the electrical resistance changes with temperature. We are free to define temperatures measured by such a thermometer in kelvins (K) to be directly proportional to the resistance \(R\), measured in ohms \((\Omega) .\) A certain resistance thermometer is found to have a resistance \(R\) of \(90.35 \Omega\) when its bulb is placed in water at the triple-point temperature \((273.16 \mathrm{~K}) .\) What temperature is indicated by the thermometer if the bulb is placed in an environment such that its resistance is \(96.28 \Omega ?\)
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