The purpose of this study is to determine if administering methadone to mechanically ventilated patients in the medical intensive care unit (ICU) requiring continuous infusions of sedatives and analgesics will decrease the time of mechanical ventilation, when initiated within 48 hours of their admission. Patients meeting enrollment criteria will be randomly assigned to receive methadone or placebo in addition to standard care. Methadone is a long acting pain medication that is approved by the Food and Drug Administration (FDA) to manage withdrawal from opioids and moderate to severe pain. Both of these indications are a frequent concern for critically ill patients that require mechanical ventilation. These patients often require intravenous (IV) opioids to manage the pain they experience due to their illness, procedures, and mechanical ventilation. During this time patients can develop physical dependence, which leads to withdrawal symptoms when the opioids are stopped or the dose is reduced. These symptoms can include agitation, pain, diarrhea and several others. Currently this is managed by a slow reduction in the dose of the IV opioid, but this can lead to prolonged time on mechanical ventilation, which has been associated with increased morbidity. Administering oral methadone to patients experiencing withdrawal symptoms has been shown to reduce and even eliminate these symptoms in the outpatient setting. This should also benefit patients in the ICU experiencing withdrawal from intravenous opioids required during their stay. It may allow for the other opioids to be discontinued more quickly, allowing for a shorter duration of mechanical ventilation.