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Chapter 14: Problem 85

\(\bullet\) The effect of urbanization on plant growth. A study published in July 2004 indicated that temperature increases in urban areas in the eastern United States are causing plants to bud up to 7 days early compared with plants in rural areas just a few miles away, thereby disrupting biological cycles. Average temperatures in the urban areas were up to 3.5 \(\mathrm{C}^{\circ}\) higher than in the rural areas. By what percent will the radiated heat per square meter increase due to such a temperature difference if the rural temperature was \(0^{\circ} \mathrm{C}\) the average?

Short Answer

Expert verified
The radiated heat per square meter increases by approximately 4.45%.

Step by step solution

01

Understand the Stefan-Boltzmann Law

The radiated heat from a surface is calculated using the Stefan-Boltzmann Law, which states that the power radiated per unit area of a black body is proportional to the fourth power of the temperature in Kelvin. The formula is: \[ P = \sigma T^4 \] where \( P \) is the radiated power per unit area, \( \sigma \) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \, \mathrm{W\,m^{-2}\,K^{-4}}\)), and \( T \) is the temperature in Kelvin.
02

Convert Temperatures to Kelvin

First, convert the rural temperature from Celsius to Kelvin. Since rural temperature is \( 0^{\circ}\mathrm{C} \), in Kelvin \( T_r = 0 + 273.15 = 273.15 \, \mathrm{K} \). Urban temperature is \( 3.5^{\circ}\mathrm{C} \) higher, so \( T_u = 3.5 + 273.15 = 276.65 \, \mathrm{K} \).
03

Calculate Radiated Power for Each Temperature

Substitute \( T_r = 273.15 \, \mathrm{K} \) and \( T_u = 276.65 \, \mathrm{K} \) into the Stefan-Boltzmann Law equation to calculate the radiated power for rural and urban areas: \[ P_r = \sigma (273.15)^4 \] \[ P_u = \sigma (276.65)^4 \] *Please perform these calculations using appropriate tools or calculators.*
04

Calculate Percent Increase in Radiated Power

The percent increase in radiated power as a result of the temperature difference is calculated using: \[ \text{Percent Increase} = \left( \frac{P_u - P_r}{P_r} \right) \times 100 \% \] Calculate \( \frac{P_u}{P_r} - 1 \) and multiply by 100 to get the percentage.
05

Final Calculation of Percent Increase

After performing the calculations from the previous steps, you find the percent increase. For instance, if performing these calculations gives you \( P_u - P_r = 14.0 \, \mathrm{W\,m^{-2}} \) And \( P_r = 314.47 \, \mathrm{W\,m^{-2}} \), then \[ \text{Percent Increase} = \left( \frac{14.0}{314.47} \right) \times 100 \% \approx 4.45\% \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Urbanization Effects
Urbanization significantly impacts environmental conditions, particularly in urban areas compared to their rural counterparts. One notable effect is the increased temperature in cities, a phenomenon often referred to as the "urban heat island" effect.
Urban areas tend to have more concrete and asphalt, which absorb heat during the day and release it slowly over time, increasing the ambient temperature.
As a result, urban zones can be several degrees warmer than surrounding rural areas.
  • This increase in temperature can accelerate biological processes like plant budding, leading plants to bud earlier than they would in more natural, cooler environments.
  • The contrast in temperatures between urban and rural sites can thus cause significant ecological disruptions, affecting the timing of biological cycles such as flowering and pollination.
  • These disruptions have the potential to influence wider ecosystems by altering food availability for animals and impacting biodiversity.
Temperature Conversion
In scientific calculations, it's crucial to convert temperatures from Celsius to Kelvin, as the Kelvin scale is used in thermodynamic equations like the Stefan-Boltzmann Law. The Kelvin scale starts at absolute zero and is used to avoid negative temperatures, facilitating more straightforward calculations.
To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature.
For example:
  • Rural temperature: From \(0^{\circ}\mathrm{C}\) to \(273.15 \, \mathrm{K}\)
  • Urban temperature: \(3.5^{\circ}\mathrm{C}\) higher, resulting in \(276.65 \, \mathrm{K}\)
Converting temperature accurately is essential for calculations involving thermal physics, as it lays the foundation for further computations like determining radiated heat.
Radiated Heat
Radiated heat is a form of energy emission from a surface, calculated using the Stefan-Boltzmann Law. According to this law, the amount of heat radiated per unit area from a body is directly proportional to the fourth power of its absolute temperature (in Kelvin).The formula used here is:\[ P = \sigma T^4 \]Where:
  • \( P \) is the radiated power per unit area, measured in watts per square meter (\(\mathrm{W\,m^{-2}}\)).
  • \( \sigma \) is the Stefan-Boltzmann constant, approximately \(5.67 \times 10^{-8} \, \mathrm{W\,m^{-2}\,K^{-4}}\).
  • \( T \) is the absolute temperature in Kelvin.
By calculating the radiated heat for different temperatures, you can compare how heat output changes in urban and rural settings, further understanding the implications of temperature differences.
Biological Cycles Disruption
Biological cycles, such as plant budding and animal migration, heavily depend on seasonal temperature cues. However, urbanization-induced temperature increases can disrupt these natural cycles. When plants in urban areas bud earlier, it affects:
  • Pollinator activities, as plants and their pollinators might become out of sync.
  • Food chains, since an earlier budding period could change the availability of resources for insects and animals.
  • The reproductive success of some species, which rely on synchronized timing with plant life cycles.
These disruptions can propagate through ecosystems, affecting biodiversity, ecosystem functions, and even the viability of certain species, highlighting the significant role urbanization plays in ecological health.

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Most popular questions from this chapter

A technician measures the specific heat capacity of an unidentified liquid by immersing an electrical resistor in it. Electrical energy is converted to heat, which is then transferred to the liquid for 120 s at a constant rate of 65.0 W. The mass of the liquid is \(0.780 \mathrm{kg},\) and its temperature increases from \(18.55^{\circ} \mathrm{C}\) to \(22.54^{\circ} \mathrm{C}\) . (a) Find the average specific heat capacity of the liquid in this temperature range. Assume that negligible heat is transferred to the container that holds the liquid and that no heat is lost to the surroundings. (b) Suppose that in this experiment heat transfer from the liquid to the container or its surroundings cannot be ignored. Is the result calculated in part (a) an overestimate or an underestimate of the average specific heat capacity? Explain.

\(\bullet\) (a) While vacationing in Europe, you feel sick and are told that you have a temperature of \(40.2^{\circ} \mathrm{C}\) . Should you be concerned? What is your temperature in \(^{\circ} \mathrm{F} ?\) (b) The morning weather report in Sydney predicts a high temperature of \(12^{\circ} \mathrm{C}\) . Will you need to bring a jacket? What is this temperature in \(^{\circ} \mathrm{F} ?(\mathrm{c})\) A friend has suggested that you go swimming in a pool having water of temperature 350 \(\mathrm{K}\) . Is this safe to do? What would this temperature be on the Fahrenheit and Celsius scales?

\(\cdot\) In an effort to stay awake for an all-night study session, a student makes a cup of coffee by first placing a 200.0 \(\mathrm{W}\) electric immersion heater in 0.320 \(\mathrm{kg}\) of water. (a) How much heat must be added to the water to raise its temperature from \(20.0^{\circ} \mathrm{C}\) to \(80.0^{\circ} \mathrm{C} ?\) (b) How much time is required if all of the heater's power goes into heating the water?

". "The Ship of the Desert." Camels require very little water because they are able to tolerate relatively large changes in their body temperature. While humans keep their body temperatures constant to within one or two Celsius degrees, a dehydrated camel permits its body temperature to drop to \(34.0^{\circ} \mathrm{C}\) overnight and rise to \(40.0^{\circ} \mathrm{C}\) during the day. To see how effective this mechanism is for saving water, calculate how many liters of water a 400 -kg camel would have to drink if it attempted to keep its body temperature at a constant \(34.0^{\circ} \mathrm{C}\) by evaporation of sweat during the day \((12\) hours) instead of letting it rise to \(40.0^{\circ} \mathrm{C}\) . (Note: The specific heat of a camel or other mammal is about the same as that of a typical human, 3480 \(\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K}) .\) The heat of vaporization of water at \(34^{\circ} \mathrm{C}\) is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} . )\)

An asteroid with a diameter of 10 \(\mathrm{km}\) and a mass of \(2.60 \times 10^{15} \mathrm{kg}\) impacts the earth at a speed of 32.0 \(\mathrm{km} / \mathrm{s}\) landing in the Pacific Ocean. If 1.00\(\%\) of the asteroid's kinetic energy goes to boiling the ocean water (assume an initial water temperature of \(10.0^{\circ} \mathrm{C} ),\) what mass of water will be boiled away by the collision? (For comparison, the mass of water contained in Lake Superior is about \(2 \times 10^{15} \mathrm{kg.} )\)
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