Question

A birch tree loses \(618 \mathrm{mg}\) of water per minute through transpiration (evaporation of water through stomatal pores). What is the rate of heat lost through transpiration?

Short answer

Answer: The rate of heat lost through transpiration is approximately 23.175 J s^-1 (Joules per second).

Step-by-step solution

Convert mass loss to kg per minute

To use the formula for heat, we need to convert the mass loss from mg to kg. Since there are \(1,000,000 \mathrm{mg}\) in a kilogram, we can convert \(618\) mg as follows: $$618 \, \mathrm{mg} = \frac{618}{1,000,000} \, \mathrm{kg} = 6.18 \times 10^{-4} \, \mathrm{kg}$$ Therefore, the mass loss of water per minute in kg is \(6.18 \times 10^{-4} \, \mathrm{kg}\).

Calculate the heat of evaporation

Now we can multiply the mass loss by the latent heat of vaporization of water, which is \(2.25 \times 10^6 \, \mathrm{J\,kg^{-1}}\), to find the heat of evaporation, \(Q\): $$Q = m L_v = (6.18 \times 10^{-4} \, \mathrm{kg}) (2.25 \times 10^6 \, \mathrm{J\,kg^{-1}}) = 1390.5 \, \mathrm{J}$$ Thus, the heat of evaporation is \(1390.5 \, \mathrm{J}\).

Calculate the rate of heat loss

Finally, we can divide the heat of evaporation by the time (in minutes) to find the rate of heat loss: $$\mathrm{Rate\, of\, heat\, loss} = \frac{Q}{\mathrm{time}} = \frac{1390.5 \, \mathrm{J}}{1\,\mathrm{min}}$$ As there are \(60\) seconds in a minute, we convert the heat loss rate to Joules per second: $$\mathrm{Rate\, of\, heat\, loss} = \frac{1390.5 \, \mathrm{J}}{60\,\mathrm{s}} = 23.175 \, \mathrm{J\,s^{-1}}$$ Hence, the rate of heat lost through transpiration is approximately \(23.175 \, \mathrm{J\,s^{-1}}\) (Joules per second).