Blood is a buffer solution. When carbon dioxide is absorbed into the bloodstream, it produces carbonic acid and lowers the pH . The body compensates by producing bicarbonate, a weak base to partially neutralize the acid. The equation which models blood pH in this situation is \( p H(x)=6.1+\log \left(\frac{800}{x}\right) \) where \( x \) is the partial pressure of carbon dioxide in arterial blood, measured in torr. Find the partial pressure of carbon dioxide in arterial blood if the pH is 7.8 .

To find the partial pressure of carbon dioxide \( x \) in arterial blood when the pH is 7.8, we can start by substituting \( pH = 7.8 \) into the given equation: \[ pH(x) = 6.1 + \log\left(\frac{800}{x}\right) \] Substituting \( 7.8 \) for \( pH(x) \): \[ 7.8 = 6.1 + \log\left(\frac{800}{x}\right) \] Next, we will isolate the logarithmic term: \[ 7.8 - 6.1 = \log\left(\frac{800}{x}\right) \] Calculating the left side: \[ 1.7 = \log\left(\frac{800}{x}\right) \] Now, we will convert the logarithmic equation to its exponential form: \[ \frac{800}{x} = 10^{1.7} \] Next, we will calculate \( 10^{1.7} \): \[ 10^{1.7} \approx 50.1187 \] Now we can set up the equation: \[ \frac{800}{x} = 50.1187 \] To find \( x \), we can rearrange the equation: \[ x = \frac{800}{50.1187} \] Now, we will calculate \( x \). Calculate the value by following steps: - step0: Calculate: \(\frac{800}{50.1187}\) - step1: Convert the expressions: \(\frac{800}{\frac{501187}{10000}}\) - step2: Multiply by the reciprocal: \(800\times \frac{10000}{501187}\) - step3: Multiply: \(\frac{800\times 10000}{501187}\) - step4: Multiply: \(\frac{8000000}{501187}\) Calculating the value of \( x \): \[ x \approx 15.96 \] Thus, the partial pressure of carbon dioxide in arterial blood when the pH is 7.8 is approximately \( 15.96 \) torr.