Problem 31 - University Physics with Modern Physics

Question

BIO (a) If a chest x ray delivers 0.25 \(\mathrm{mSv}\) to 5.0 \(\mathrm{kg}\) of tissue, how many total joules of energy does this tissue receive? (b) Natural radiation and cosmic rays deliver about 0.10 \(\mathrm{mSv}\) per year at sea level. Assuming an \(\mathrm{RBE}\) of \(1,\) how many rem and rads is this dose, and how many joules of energy does a 75 -kg person receive in a year? (c) How many chest xays like the one in part (a) would it take to deliver the same total amount of energy to a 75 -kg person as she receives from natural radiation in a year at sea level, as described in part (b)?

Step-by-Step Solution

Answer

(a) 1.25 J, (b) 0.01 rem, 0.01 rads, 7.5 J, (c) 6 x-rays.

Step 1: Understand the Units

The problem involves different units: sieverts (Sv) and rems for radiation dose, and joules for energy. We need to convert sieverts to joules for the energy calculation. Note that 1 mSv = 0.001 Sv.

Step 2: Calculate Energy Received from the X-Ray

Radiation dose in sieverts (Sv) is the energy deposited per kilogram. For the chest x-ray, the dose is 0.25 mSv to 5.0 kg. First convert mSv to Sv:\[ 0.25 \text{ mSv} = 0.25 \times 0.001 \text{ Sv} = 0.00025 \text{ Sv} \]Then calculate the energy deposited:\[ E = \text{Dose (Sv)} \times \text{Mass (kg)} \times 1000 \text{ J/kg} = 0.00025 \times 5.0 \times 1000 = 1.25 \text{ J} \]

Step 3: Calculate Dose and Energy from Natural Radiation

For natural radiation, the dose is 0.10 mSv per year. First convert this to Sv:\[ 0.10 \text{ mSv} = 0.10 \times 0.001 \text{ Sv} = 0.0001 \text{ Sv} \]RBE is 1, so rems = sieverts \(\times 100\). Hence:\[ \text{Dose in rem} = 0.0001 \times 100 = 0.01 \text{ rem} \]To calculate rads, note that 1 rad = 0.01 Joules/kg. Since RBE is 1, rems = rads. So:\[ \text{Dose in rads} = 0.01 \text{ rads} \]Finally, calculate energy for a 75 kg person:\[ E = \text{Dose (Sv)} \times \text{Mass (kg)} \times 1000 \text{ J/kg} = 0.0001 \times 75 \times 1000 = 7.5 \text{ J} \]

Step 4: Calculate the Number of Equivalent X-Rays

The energy from one chest x-ray is 1.25 J. To find the number of x-rays that match the energy from natural radiation (7.5 J):\[ n = \frac{\text{Total energy from natural radiation}}{\text{Energy per x-ray}} = \frac{7.5}{1.25} = 6 \]

Key Concepts

Sieverts to Joules conversion

Radiation dose is often measured in sieverts (Sv), which indicates the potential harm a person might receive from exposure to various forms of radiation. However, when examining the energy deposited in tissue, it can be more intuitive to work with joules (J) – the standard unit of energy in physics. To transition from sieverts to joules, one needs to follow a straightforward approach:- Recognize that 1 millisievert (mSv) equals 0.001 Sv, as milli- denotes one-thousandth.- The formula to convert is straightforward: \[ E = \text{Dose (Sv)} \times \text{Mass (kg)} \times 1000 \, \text{J/kg} \] - Here, 'Dose' is the radiation dose in sieverts, and 'Mass' is the mass of the tissue in kilograms. - The constant 1000 J/kg helps make the conversion clear because it represents the energy in joules deposited per kilogram of tissue for each sievert.- For instance, if 5 kg of tissue receives a dose of 0.00025 Sv (from a chest x-ray), the total energy would be:\[ E = 0.00025 \times 5 \times 1000 = 1.25 \, \text{J} \]This method ensures accurate conversion from a biological dose to a tangible energy form, offering a clearer picture of the actual impact of radiation exposure.

Chest X-ray energy calculation

The calculation of energy received from a chest X-ray involves understanding how radiation dose impacts the human tissue, measured in sieverts. By understanding the specific radiation dose parameters:- A typical chest X-ray may deliver around 0.25 mSv to the tissue.- To calculate the energy imparted, follow these steps: 1. Convert millisieverts to sieverts: \[ 0.25 \, \text{mSv} = 0.25 \times 0.001 = 0.00025 \, \text{Sv} \] 2. Use the converted dose to find energy in joules using the formula: \[ E = \text{Dose (Sv)} \times \text{Mass (kg)} \times 1000 \, \text{J/kg} \] - For 5 kg of tissue, \[ E = 0.00025 \times 5 \times 1000 = 1.25 \, \text{J} \]- This calculation helps visualize the small amount of energy transferred during a routine imaging process, emphasizing the low yet significant radiation impact on tissues.Unlike everyday exposures, X-rays focus specifically, allowing precise medical insights with minimal energy implications comparatively.

Natural radiation exposure

Natural radiation exposure, also known as background radiation, affects everyone. It originates from cosmic rays, the earth's crust, and even within our own bodies. The annual dose from natural sources at sea level is generally about 0.10 mSv.To understand the impact:- Convert millisieverts to sieverts: \[ 0.10 \, \text{mSv} = 0.10 \times 0.001 = 0.0001 \, \text{Sv} \]- Determine the biological equivalent dose using an RBE (relative biological effectiveness) of 1. Hence, rems equal sieverts times 100: \[\text{Dose in rem} = 0.0001 \times 100 = 0.01 \, \text{rem} \]- For a 75 kg person, the energy received annually can be computed: \[ E = 0.0001 \times 75 \times 1000 = 7.5 \, \text{J} \]- Translate to rads considering that 1 rad equals 0.01 J/kg: \[\text{Dose in rads} = 0.01 \, \text{rads} \]This natural radiation is unavoidable, indicating the small yet continuous exposure every individual undergoes yearly. It’s the baseline from which humans gauge additional environmental radiation and safety levels.